  Let $G$ be the semidirect product $\Gamma \rtimes F_2$ where $\Gamma$ is
either the free group $F_n$, $n > 1$ or the fundamental group $S_g$ of a closed
surface of genus $g > 1$. We prove that $G$ is incoherent, solving two problems
posed by D. Wise. This implies an affirmative answer to a question of J.
Hillman on the fundamental group of a surface bundle over a surface. Although
many groups have been shown to be incoherent using virtual algebraic fibering,
we also show that not every free-by-free group virtually algebraically fibers.
