As far as things like run/run_until_complete/etc go, I think doctests should have the same semantics as async REPLs, whatever those end up being.

Given that we don't actually have mature async REPLs, that the core feature to enable them only landed a few days ago, and the 3.8 freeze is almost here, I think we should defer the doctest discussion for 3.9. Hopefully by then, things will have settled down more in async REPL land, and it'll be more obvious what doctest should do.