Mask all URIs so only base domain name is shown in address bar

You can't control the uri displayed in the address bar, unless you redirect the request (on the client side) to /.

Then you would have to do some real complex irule processing to hold the entire server response at the BigIP, and then get the client to collect it, using a redirect to / with some sort of token to trigger the actual response.

I can see how it could work, but it's really, really hard.

That's why you cannot find a definitive answer ...