Is there any reason to use WebSockets on native apps other than if you want to share one server between native and web?
That aside, it seems a bit ridiculous, given that we already have sockets, to bolt on a protocol that exists almost solely to work around security issues involved in a browser opening a socket.
If you want bidirectional communication to your app, it's much easier to make a server that uses the WebSocket Protocol (tornado, etc.) instead of rolling your own server and protocol.