Well, you can't really call it an F(flat5) or any terminological variation thereof. Because that would imply that you've taken the 5th and flattened it. But in this case, a flat-five has been added, but the "normal" five is still there, unchanged. Really, that B note is more like a raised 4 (or #11) because of that.
As far as what "goes" with it, #11 chords suggest the lydian mode (I've even heard them called "lydian" chords sometimes). Since F lydian is the same as C ionion, which is the same as a C major scale, then any chord that's diatonic to the key of C would work with this chord: C, Dm, Em, F (normal), G or G7, Am, Bdim.
JJ