It doesn't seem impossible, as long as we're not talking 100% perfect anonymity nor 100% verifiability.
Perhaps make users upload receipt dated within last x days, scratching out some details, and writing their username on it. It would take an admin all of 5 seconds to review and click approve, which would delete the receipt and allow the review through. Perhaps after x amount of verified reviews, just whitelisting the user. So at this point, your IP, username, and date of purchase are known to the review system, so not 100% anonymous.
Reviews without a receipt, or in waiting, are allowed but flagged as unverified or unsubstantiated. Probably allow them to be viewed, but not by default.
And of course, your typical spam preventions, review bombing deletion, etc.
All this will do is make it painful enough for regular users to never use the service (who even gets receipts from a restaurant anymore?) while malicious ones will be able to generate fake ones in seconds to review bomb.
Perhaps make users upload receipt dated within last x days, scratching out some details, and writing their username on it. It would take an admin all of 5 seconds to review and click approve, which would delete the receipt and allow the review through. Perhaps after x amount of verified reviews, just whitelisting the user. So at this point, your IP, username, and date of purchase are known to the review system, so not 100% anonymous.
Reviews without a receipt, or in waiting, are allowed but flagged as unverified or unsubstantiated. Probably allow them to be viewed, but not by default.
And of course, your typical spam preventions, review bombing deletion, etc.