  Let $G = H_1 * ... * H_k * F_r$ be a torsion-free group and $\phi$ an
automorphism of $G$ that preserves this free factor system. We show that when
$\phi$ is fully irreducible and atoroidal relative to this free factor system,
the mapping torus $\Gamma = G \rtimes_{\phi} \mathbb{Z}$ acts relatively
geometrically on a hyperbolic CAT(0) cube complex. This is a generalisation of
a result of Hagen and Wise for hyperbolic free-by-cyclic groups.
