Green threads / coroutines / fibers are isomorphic with async keyword transparently implemented as a continuation passing style transform, which is how async callbacks usually work. Actual CPU-style stacks in a green thread scenario are nested closure activation records in an explicit continuation passing style scenario, and are implicit closure activation records (but look like stacks) when using an 'async' compiler-implemented CPS.

Properly composed awaits (where each function entered is entered via an await) build a linked list of activation records in the continuations as they drill down. This linked list is the same as the stack (i.e. serves the same purpose and contains the same data in slightly different layout) in a green threads scenario.

What makes all these things different is how much they expose the underlying mechanics, and the metaphors they use in that exposition. But they're not orthogonal.

(If you meant 'async' as in async IO explicitly, rather than the async / await keyword with CPS transform as implemented in C#, Python, Javascript, etc., then apologies.)