Would a scheme like the following work to allow authentication of a sender:
A nonce is appended to data, and the result is encrypted. The encrypted pack is sent to the other party. The sender then contacts the receiver through an authenticated channel (or one that is sufficiently good for the current purposes) and provides them with the nonce. When the receiver gets the encrypted file, they decrypt it. If the nonce matches the one given through the secondary channel, the encrypted pack is known to be from the sender.
Now, I realize no such scheme could provide non-repudiation, but the purpose in this case is simply to ensure that the encrypted pack has not been tampered with in transit. My presumption is that tampering with the encrypted pack without decrypting it has a high probability of creating gibberish, so the main threat of an MITM is the pack being removed and replaced with something else.
Does the approach above sound reasonable?