I just stumbled upon a Reddit post titled:

My wife and I's seafood collaboration dinner. How does it look?

Sure enough, the top comment immediately points out that it should be "my wife's and my". However, a cross-post to the Grammar subreddit produced the following comment:

It's fine as it is written. "my wife and I" is a noun phrase, functioning as a subjective pronoun in the singular and made possessive with the apostrophe. It is exactly the same as "our".

It seems weird because you would never use "I's" on its own but it is not on its own here - it is part of a noun phrase.

That's a rather intriguing argument. Does it hold any water?