  We show that if $G$ is a group of type $FP_{n+1}^{\mathbb{Z}_2}$ that is
coarsely separated into three essential, coarse disjoint, coarse complementary
components by a coarse $PD_n^{\mathbb{Z}_2}$ space $W,$ then $W$ is at finite
Hausdorff distance from a subgroup $H$ of $G$; moreover, $G$ splits over a
subgroup commensurable to a subgroup of $H$. We use this to deduce that
splittings of the form $G=A*_HB$, where $G$ is of type
$FP_{n+1}^{\mathbb{Z}_2}$ and $H$ is a coarse $PD_n^{\mathbb{Z}_2}$ group such
that both $|\mathrm{Comm}_A(H): H|$ and $|\mathrm{Comm}_B(H): H|$ are greater
than two, are invariant under quasi-isometry.
