Assume that Alice has a file $F$ which she is going to send, in encrypted form to Bob. Alice possesses $F$ and an encryption key $K$. She sends to Bob the encryption of $F$ using $K$, $E(F,K)$ as well as a compact message authentication code which could be the hash of the file $H = {\rm hash}(F)$, which is used as a unique blinded identifier of the file. It's assumed that multiple users may have the same file, encrypted with different keys, and we wish to detect that they are the same by virtue of $H$ being the same, but Bob needs to know that $H$ matches $F$ without having $F$ unencrypted.
Does there exist any kind of proof that the hash $H$ (or other compact MAC) corresponds to the file $F$? The hashing/MAC algorithm and encryption algorithm (symmetric, asymmetric, homomorphic) are secondary to being able to prove this.