Right: and that's not a single, continuous function that works for all inputs—specifically, it fails on the input B itself. BxB = 0.

Any solution will have a discontinuity in its output vector angles. I don't know how this problem is applied in computer graphics, but you probably want to avoid rendering objects in the vicinity of a discontinuity: you'd get some kind of flickering artifact when you cross it, with small ɛ-displacements being amplified into something much larger.