Rewrite header "WWW-Authenticate: Basic realm="

What I'm saying though is that the browser will most likely not handle this non-standard 401 the way you're expecting. As a test, try this on a simple VIP:

when HTTP_REQUEST { 
    if { not ( [HTTP::header exists Authorization] ) } {
        HTTP::respond 401 WWW-Authenticate "Xbasic realm=\"TEST\""
    }
}

What does the browser do? In all of my testing the browser does absolutely nothing. It just stalls.

So if there's no authentication on the backend, and apparently no authentication on the frontend, what are you trying to do?