> if you have several tabs, you need a SharedWorker to share the connection between the tabs.

You don't have to use a SharedWorker, you can also do domain sharding. Since the concurrent connection limit is per domain, you can add a bunch of DNS records like SSE1.example.org -> 2001:db8::f00; SSE2.example.org -> 2001:db8::f00; SSE3.example.org -> 2001:db8::f00; and so on. Then it's just a matter of picking a domain at random on each page load. A couple hundred tabs ought to be enough for anyone ;)