From my side, I'm looking to experimentally build a network testing tool that will need to speak a fair few protocols, both classic tcp and multicast-based,
Tornado is a _Web_ framework. A requirement for UDP, and multicast UDP, narrows suitable choices to Twisted and asyncio.
Twisted is around much longer, and also has a broader focus .. lots of protocols implemented on top of TCP/IP.
Asyncio is opiniated towards a co-routine based style. Twisted probably more towards "classical" Deferred based style, though you can go co-routine style as well.
and have a web api living on top of it that most likely will have a websocket for pumping data to the browser. It'll also need to write out JUnit-compatible
If you need WebSocket anyway, you can directly compare Twisted and asyncio programming with these examples:
Disclaimer: I am affiliated with that stuff.
I'd like to be able to serve the rest of the web api using a pyramid wsgi app if
Twisted can act as a WSGI host, and that can be combined with WebSocket also (in one server, and both services running on one port).