Incoherence of free-by-free and surface-by-free groups

Let $G$ be the semidirect product $\mathrm{\Gamma }⋊F_2$ where $\mathrm{\Gamma }$ 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.