Warbird Information Exchange

Can a true flying wing (no booms with empennage), or nuttin', just a wing, with small blisters allowed for cabin and cockpit, but that's it, exceed the speed of sound? I'm talking level flight. Failing that, then in a dive. For some reason, in my minds eye, I see this as an impossibility due to what I imagine to be severe controllability issues - mach tuck, uncommanded and uncontrollable pitch-up, lack of control authority, etc. I fail to see even a quad-redundant DFBW flight control system pulling that rabbit out of the hat. But maybe it has already been achieved and someone here can educate me on the why, when, where and how of it all. Another problem to achieving supersonic flight that I foresee is the use of immensely thick airfoils which such aircraft seem to require, creating mucho, serious drag. If that is a legit issue, then can a thin-wing flying wing variant pull the trick off (although sacrificing it's raison d'etre, that of load carrying ability)? Believe it or not, this thought has been noodling it's way thru my brain for quite sometime, because it appears that all FW's that I'm aware of are subsonic.

Thanks for any and all thoughtful input.