Structure of quasiconvex virtual joins

Let $G$ be a relatively hyperbolic group and let $Q$ and $R$ be relatively quasiconvex subgroups. It is known that there are many pairs of finite index subgroups $Q^{\mathrm{\prime }}{⩽}_{f}Q$ and $R^{\mathrm{\prime }}{⩽}_{f}R$ such that the subgroup join $⟨Q^{\mathrm{\prime }},R^{\mathrm{\prime }}⟩$ is also relatively quasiconvex, given suitable assumptions on the profinite topology of $G$. We show that the intersections of such joins with maximal parabolic subgroups of $G$ are themselves joins of intersections of the factor subgroups $Q^{\mathrm{\prime }}$ and $R^{\mathrm{\prime }}$ with maximal parabolic subgroups of $G$. As a consequence, we show that quasiconvex subgroups whose parabolic subgroups are almost compatible have finite index subgroups whose parabolic subgroups are compatible, and provide a combination theorem for such subgroups.