Other people who are better with these things could probably conceptualize it better than me.

What I'd like to be able to do is declare a trait that says "this thing can yield the following patterns, can destructure in this pattern, and these are the functions you'd call to extract references to those values"

And then be able to use match on it as usual.

Putting it another way, I wish an algebraic data type "enum" was something any struct could "be", with the type system providing the means to express that that's the case, and the programmer being able to provide the "how" and with existing Rust enums merely being the default way of doing this.

Haskell and F# have this feature. In F# it is called active patterns. In Haskell the feature is pattern synonyms (maybe combined with view patterns) and is used to e.g. pattern match on the prefix or suffix of a sequence-like structure, or to match a tree as a an element and left/right node without the caller seeing a bunch of other tree metadata.

Interesting idea. I feel a bit conflicted about it, but it's probably something I'd have to think about.

I suppose it's possible to somewhat approximate this in existing Rust by creating a trait with a function that returns a destructure-able type, performing the transformations that you would have implemented in your hypothetical trait, then pattern match over that. Not sure if that would be "good enough".