  Let $\phi:G\to G$ be an automorphism of a group which is a free-product of
finitely many groups each of which is freely indecomposable and two of the
factors contain proper finite index characteristic subgroups. We show that $G$
has infinitely many $\phi$-twisted conjugacy classes. As an application, we
show that if $G$ is the fundamental group of a three-manifold that is not
irreducible, then $G$ has property $R_\infty$, that is, there are infinitely
many $\phi$-twisted conjugacy classes in $G$ for every automorphism $\phi$ of
$G$.
