Let $G=G_{1}\ast \dots \ast G_{k}\ast F$ be a countable group which splits as a free product, where all groups $G_{i}$ are freely indecomposable and not isomorphic to ℤ, and $F$ is a finitely generated free group. If for all $i\in \{1,\dots ,k\}$ , both $G_{i}$ and its outer automorphism group $\text{Out}(G_{i})$ satisfy the Tits alternative, then $\text{Out}(G)$ satisfies the Tits alternative. As an application, we prove that the Tits alternative holds for outer automorphism groups of right-angled Artin groups, and of torsion-free groups that are hyperbolic relative to a finite family of virtually polycyclic groups.
