Non-Redemption Agreement, dated July 9, 2023

Exhibit 10.2

Execution Version

NON-REDEMPTION AGREEMENT

This NON-REDEMPTION AGREEMENT (this “Agreement”), dated as of July 9, 2023, is made by and among Alpha Healthcare Acquisition Corp. III, a Delaware corporation (the “Company” or “ALPA”), and the Backstop Investor (as defined below).

WHEREAS, the Company is a special purpose acquisition company whose Class A Common Stock (“Common Stock”) is traded on The Nasdaq Capital Market under the symbol “ALPA”, and whose warrants (“Warrants”) are traded on The Nasdaq Capital Market under the symbol “ALPAW”, among other securities of the Company;

WHEREAS, the Company, Candy Merger Sub, Inc., a Delaware corporation and wholly owned subsidiary of ALPA (“Merger Sub”), and Carmell Therapeutics Corporation (“Carmell”), have entered into a Transaction Agreement, dated as of January 4, 2023 (as may be amended from time to time, the “Transaction Agreement”), pursuant to which Merger Sub will merge with and into Carmell, with Carmell surviving as a wholly owned subsidiary of ALPA (such transaction, the “Business Combination”);

WHEREAS, the Company and the Backstop Investor on behalf of certain funds, investors, entities or accounts that are managed, sponsored or advised by the Backstop Investor or its affiliates (the “Backstop Investor”) are entering into this Agreement in anticipation of the closing of the Business Combination contemplated by the Transaction Agreement;

WHEREAS, pursuant to the Company’s Second Amended and Restated Certificate of Incorporation, effective March 22, 2021 (the “COI”), in its capacity as a holder of Common Stock, the Backstop Investor has the right to require that the Company redeem the Backstop Investor Shares held by the Backstop Investor in connection with the Business Combination, for the Redemption Price (as defined in the COI), representing the right to receive the Backstop Investor’s portion of the funds currently in the Company’s trust account, to the extent the Backstop Investor exercises such redemption right;

WHEREAS, the Company filed a definitive merger proxy, dated June 23, 2023, with a deadline to exercise the redemption rights of shares of Common Stock of 5:00 p.m., local time, on July 7, 2023 (two Business Days before the special meeting (the “Meeting”) of stockholders of the Company to approve the Business Combination) (the “Redemption Deadline”);

WHEREAS, pursuant to the terms of this Agreement, if a redemption was previously submitted by the Backstop Investor for the number of shares of Common Stock as set forth in Exhibit A hereto (the “Backstop Investor Shares”), the Backstop Investor desires to reverse the redemption request solely with respect to the Backstop Investor Shares subject to this Agreement; for the avoidance of doubt, the Backstop Investor may have voting and investment power over additional shares of Common Stock which will not be subject to this Agreement (such shares, “Non-Backstop Investor Shares”) and the Backstop Investor retains the right in its sole discretion not to reverse any redemption request previously submitted with respect to any such Non-Backstop Investor Shares; and

WHEREAS, all capitalized terms used but not defined herein shall have the respective meanings specified in the Transaction Agreement.

NOW, THEREFORE, in consideration of the mutual agreements set forth herein, the parties agree as follows:

1.    Non-Redemption Agreement. Subject to the conditions set forth in this Agreement, the Backstop Investor irrevocably and unconditionally hereby agrees that, in the event the Backstop Investor has previously elected to redeem, tendered or submitted any of the Backstop Investor Shares for redemption in connection with the Business Combination, the Backstop Investor shall rescind or reverse any such redemption no later than the closing of the Business Combination.

2.    Representations and Warranties. Each of the parties hereto represents and warrants to the other party that: (a) it is a validly existing company, partnership or corporation, in good standing under the laws of the jurisdiction of its formation or incorporation; (b) this Agreement constitutes a valid and legally binding obligation on it in accordance with its terms, subject to laws relating to bankruptcy, insolvency and relief of debtors, and laws governing specific performance, injunctive relief and other equitable remedies; (c) the execution, delivery and performance of this Agreement by it has been duly authorized by all necessary corporate action, and (d) the execution, delivery and performance of this Agreement will not result in a violation of its Certificate of Formation or Certificate of Incorporation, as applicable, or conflict with, or constitute a default (or an event that with notice or lapse of time or both would become a default) under, or give to others any rights of termination, amendment, acceleration or cancellation of, any agreement or instrument to which it is a party or by which it is bound. The Backstop Investor represents and warrants to the Company, that, as of the date hereof, the Backstop Investor beneficially owns the number of shares of Common Stock set forth opposite the Backstop Investor’s name on Exhibit A hereto.

3.    Additional Covenants. The Backstop Investor hereby covenants and agrees that, except for this Agreement, the Backstop Investor shall not, at any time while this Agreement remains in effect, (i) enter into any agreement or take any action that would make any representation or warranty of Backstop Investor contained herein untrue or inaccurate in any material respect or have the effect of preventing or disabling Backstop Investor from performing any of its obligations under this Agreement, or (ii) purchase the Backstop Investor Shares at a price higher than the price offered through the Company’s redemption process.

4.    Expenses. Each party shall be responsible for its own fees and expenses related to this Agreement and the transactions contemplated hereby.

5.    Termination. This Agreement and all of its provisions shall terminate and be of no further force or effect upon the earliest to occur of (a) the termination of the Transaction Agreement in accordance with its terms, (b) the mutual written consent of the parties hereto, and (c) September 1, 2023, if the Business Combination has not been consummated by such date. Upon such termination of this Agreement, all obligations of the parties under this Agreement will terminate, without any liability or other obligation on the part of any party hereto to any person in respect hereof or the transactions contemplated hereby; provided that, notwithstanding the foregoing or anything to the contrary in this Agreement, the termination of this Agreement pursuant to clause (a) above shall not affect any liability on the part of any party for an intentional breach of this Agreement. This Section 5 through and including Section 22 of this Agreement will survive the termination of this Agreement.

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6.    Trust Account Waiver. The Backstop Investor acknowledges that the Company has established a trust account (the “Trust Account”) containing the proceeds of its initial public offering (“IPO”) and certain proceeds of a private placement (including interest accrued from time to time thereon) for the benefit of its public stockholders and certain other parties (including the underwriters of the IPO). For good and valuable consideration, the receipt and sufficiency of which are hereby acknowledged, the Backstop Investor hereby agrees (on its own behalf and on behalf of its related parties) that it does not now and shall not at any time hereafter have any right, title, interest or claim of any kind in or to any assets held in the Trust Account, and it shall not make any claim against the Trust Account, regardless of whether such claim arises as a result of, in connection with or relating in any way to this Agreement or any other matter, and regardless of whether such claim arises based on contract, tort, equity or any other theory of legal liability (any and all such claims are collectively referred to hereafter as the “Released Claims”); provided, that the Released Claims shall not include any rights or claims of the Backstop Investor or any of its related parties as a shareholder of the Company to the extent related to or arising from any Backstop Investor Shares. The Backstop Investor hereby irrevocably waives (on its own behalf and on behalf of its related parties) any Released Claims that it may have against the Trust Account now or in the future as a result of, or arising out of, this Agreement and will not seek recourse against the Trust Account with respect to the Released Claims. For the avoidance of doubt, this provision shall not restrict the Backstop Investor’s Redemption Rights (as defined in the COI) with respect to the Non-Backstop Investor Shares.

7.    Public Disclosure. The Company shall file a Current Report on Form 8-K with the SEC (the “Current Report”) reporting the material terms of this Agreement but not including the names of the Backstop Investor and its affiliates and/or advised funds, unless required by law. The Company shall not, and shall cause its representatives to not, disclose any material non-public information to the Backstop Investor concerning the Company, the Common Stock or the Business Combination, other than the existence of this Agreement, such that the Backstop Investor shall not be in possession of any such material non-public information from and after the filing of the Current Report. Notwithstanding anything in this Agreement to the contrary, the Backstop Investor agrees that the Company shall have the right to publicly disclose the nature of the Backstop Investor’s commitments, arrangements and understandings under and relating to this Agreement in any filing by the Company with the SEC.

8.    Governing Law. This Agreement, the rights and duties of the parties hereto, and any disputes (whether in contract, tort or statute) arising out of, under or in connection with this Agreement will be governed by and construed and enforced in accordance with the laws of the State of Delaware, without giving effect to its principles or rules of conflict of laws to the extent such principles or rules would require or permit the application of the laws of another jurisdiction. The parties irrevocably and unconditionally submit to the exclusive jurisdiction of the United States District Court for the District of Delaware or, if such court does not have jurisdiction, the Delaware state courts located in Wilmington, Delaware, in any action arising out of or relating to this Agreement. The parties irrevocably agree that all such claims shall be heard and determined in such a Delaware federal or state court, and that such jurisdiction of such courts with respect thereto will be exclusive. Each party hereby waives, and agrees not to assert, as a defense in any action, suit or proceeding arising out of or relating to this Agreement that it is not subject to such jurisdiction, or that such action, suit or proceeding may not be brought or is not maintainable in such courts or that the venue thereof may not be appropriate or that this Agreement may not be enforced in or by such courts. The parties hereby consent to and grant any such court jurisdiction

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over the person of such parties and over the subject matter of any such dispute and agree that mailing of process or other papers in connection with any such action, suit or proceeding in the manner provided in Section 19 hereof or in such other manner as may be permitted by law, will be valid and sufficient service thereof.

9.    Waiver of Jury Trial. To the extent not prohibited by applicable law that cannot be waived, each of the parties hereto irrevocably waives any right it may have to trial by jury in respect of any litigation based on, arising out of, under or in connection with this Agreement or any course of conduct, course of dealing, verbal or written statement or action of any party hereto or thereto, in each case, whether now existing or hereafter arising, and whether in contract, tort, statute, equity or otherwise. Each party hereby further agrees and consents that any such litigation shall be decided by court trial without a jury and that the parties to this Agreement may file a copy of this Agreement with any court as written evidence of the consent of the parties to the waiver of their right to trial by jury.

10.    Freely Tradable. The Company confirms that (i) the Backstop Investor Shares will be freely tradeable without restrictive legends following the Business Combination; (ii) the Backstop Investor Shares will not require re-registration pursuant to a registration statement filed with the SEC on Form S-1 or Form S-3 or equivalent following the Business Combination; and (iii) that the Backstop Investor shall not be identified as a statutory underwriter in any registration statement filed with the SEC on Form S-1 or Form S-3 or equivalent.

11.    Non-Reliance. The Backstop Investor has had the opportunity to consult its own advisors, including financial and tax advisors, regarding this Agreement or the arrangements contemplated hereunder and the Backstop Investor hereby acknowledges that neither the Company nor any representative or affiliate of the Company has provided or will provide the Backstop Investor with any financial, tax or other advice relating to this Agreement or the arrangements contemplated hereunder.

12.    No Third-party Beneficiaries. This Agreement shall be for the sole benefit of the parties to this Agreement, Carmell and their respective successors and permitted assigns. Except as expressly named in this Section 12, this Agreement is not intended, nor shall be construed, to give any Person, other than the parties to this Agreement, Carmell and their respective successors and assigns, any legal or equitable right, benefit or remedy of any nature whatsoever by reason this Agreement.

13.    Assignment. This Agreement and all of the provisions hereof will be binding upon and inure to the benefit of the parties hereto and their respective successors and permitted assigns. Neither this Agreement nor any of the rights, interests or obligations hereunder will be assigned (including by operation of law) without the prior written consent of the non-assigning party hereto (not to be unreasonably withheld, conditioned or delayed). Notwithstanding the foregoing, the Backstop Investor may transfer its rights, interests and obligations hereunder to one or more investment funds or accounts managed or advised by the Backstop Investor (or a related party or affiliate) and to the extent such transferee is not a party to this Agreement, such transferee shall agree to be bound by the terms hereof prior to any such transfer being effectuated.

14.    Specific Performance. The parties agree that irreparable damage may occur in the event that any of the provisions of this Agreement are not performed in accordance with their specific terms or are otherwise breached. It is accordingly agreed that monetary damages may not

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be an adequate remedy for such breach and the non-breaching party shall be entitled to seek injunctive relief, in addition to any other remedy that such party may have in law or in equity, and to enforce specifically the terms and provisions of this Agreement in the chancery court or any other state or federal court within the State of Delaware.

15.    Amendment. This Agreement may not be amended, changed, supplemented, waived or otherwise modified, except upon the execution and delivery of a written agreement executed by the parties hereto.

16.    Severability. If any provision of this Agreement is held invalid or unenforceable by any court of competent jurisdiction, the other provisions of this Agreement will remain in full force and effect. Any provision of this Agreement held invalid or unenforceable only in part or degree will remain in full force and effect to the extent not held invalid or unenforceable.

17.    No Partnership, Agency or Joint Venture. This Agreement is intended to create a contractual relationship between the Backstop Investor, on the one hand, and the Company, on the other hand, and is not intended to create, and does not create, any agency, partnership, joint venture or any like relationship between the parties.

18.    Blocker Provision. Notwithstanding anything to the contrary contained herein, the number of shares of Common Stock that may be acquired by the Backstop Investor upon any exercise of Warrants in the Company and its successor(s) shall be limited to the extent necessary to ensure that, following such exercise (or other issuance), the total number of shares of Common Stock then beneficially owned by such Backstop Investor and its affiliates and any other persons whose beneficial ownership of Common Stock would be aggregated with the Backstop Investor’s for purposes of Section 13(d) of the Exchange Act, does not exceed 9.99% of the total number of issued and outstanding shares of Common Stock (including for such purpose the shares of Common Stock issuable upon such exercise). For such purposes, beneficial ownership shall be determined in accordance with Section 13(d) of the Exchange Act and the rules and regulations promulgated thereunder. Each delivery of a Notice of Exercise hereunder will constitute a representation by the Backstop Investor that it has evaluated the limitation set forth in this paragraph and determined that issuance of the full number of Warrant Shares requested in such Notice of Exercise is permitted under this paragraph. This provision shall not restrict the number of shares of Common Stock which a Backstop Investor may receive or beneficially own in order to determine the amount of securities or other consideration that such Backstop Investor may receive in the event of a merger or other business combination or reclassification involving the Company. For the avoidance of doubt, this restriction shall not be conferred upon any third party to whom the Backstop Investor transfers such Warrants. This restriction may not be waived.

19.    Notices. All notices, consents, waivers and other communications under this Agreement must be in writing and will be deemed to have been duly given (a) if personally delivered, on the date of delivery; (b) if delivered by express courier service of national standing for next day delivery (with charges prepaid), on the Business Day following the date of delivery to such courier service; (c) if delivered by electronic mail, on the date of transmission if on a Business Day before 5:00 p.m. local time of the business address of the recipient party (otherwise on the next succeeding Business Day), provided the sender receives no bounce-back or similar message indicating non-delivery; in each case to the appropriate addresses set forth below (or to such other addresses as a party may designate by notice to the other parties in accordance with this Section 19):

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If to the Company prior to consummation of the Business Combination:

Alpha Healthcare Acquisition Corp. III

1177 Avenue of the Americas, 5th Floor

New York, New York 10036

Attention: Rajiv Shukla

E-mail: ***@***

with a copy (which will not constitute notice) to:

Goodwin Procter LLP

The New York Times Building

620 Eighth Avenue

New York, NY 10018

Attention: Jeffrey A. Letalien and Jocelyn M. Arel

Email: ***@*** and ***@***

If to the Company after consummation of the Business Combination:

Carmell Therapeutics Corporation

2403 Sidney Street, Suite 300

Pittsburgh, PA 15203

Attention: Randolph W. Hubbell

E-mail: ***@***

If to the Backstop Investor:

1200 N Federal Hwy, Ste 200

Boca Raton, FL 33432

Email: ***@***

with a copy (which will not constitute notice) to:

DLA Piper LLP (US)

555 Mission Street, Suite 2400

San Francisco, CA 94105-2933

Attention: Jeffrey C. Selman

Email: ***@***

20.    Counterparts. This Agreement may be executed in two or more counterparts (any of which may be delivered by electronic transmission), each of which shall constitute an original, and all of which taken together shall constitute one and the same instrument, and shall include images of manually executed signatures transmitted by electronic format (including, without limitation, “pdf”, “tif” or “jpg”) and other electronic signatures (including, without limitation, DocuSign and AdobeSign). The use of electronic signatures and electronic records (including, without limitation, any contract or other record created, generated, sent, communicated, received, or stored by electronic means) shall be of the same legal effect, validity and enforceability as a manually executed signature or use of a paper-based record-keeping system to the fullest extent permitted by applicable law.

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21.    Entire Agreement. This Agreement and the agreements referenced herein constitute the entire agreement and understanding of the parties hereto in respect of the subject matter hereof and supersede all prior understandings, agreements or representations by or among the parties hereto to the extent that they relate in any way to the subject matter hereof.

22.    Most Favored Nation. In the event the Company enters one or more other similar non-redemption agreements with other stockholders of the Company (such non-redemption agreements, “Other Non-Redemption Agreements”, and such other stockholders of the Company, “Other Backstop Investors”) before or after the execution of this Agreement in connection with the Meeting, the Company represents that the terms of such other similar non-redemption agreements are not materially more favorable to such Other Backstop Investors thereunder than the terms of this Agreement are in respect of the Backstop Investor. In the event that any Other Backstop Investor is afforded any such more favorable terms pursuant to such similar non-redemption agreement than the Backstop Investor, the Company shall promptly inform the Backstop Investor of such more favorable terms in writing, and the Backstop Investor shall have the right to elect to have such more favorable terms included herein, in which case the parties hereto shall promptly amend this Agreement to effect the same.

[Signature page follows]

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IN WITNESS WHEREOF, this Agreement has been duly executed by the parties hereto as of the date first above written.

EXHIBIT A