Ok I can show that [tex] F_{\mu \nu}= F_{\mu \nu}^a t^a [/tex] transforms under the adjoint representation, in the sense that one of the contracted indices a gets acted on by a matrix in the adjoint rep (which is kinda weird I think), but the proof relies on the fact that it is contracted with the generators. I cant figure out why [tex] \psi_a \psi_b [/tex] would transform under the adjoint because there are no generators kicking about, so the proof I did seems not to work.
RedX, when I said fundamental rep of AxB I guess I assumed that this would be isomorphic to a group whose fundamental rep would be defined.
