  Let $M$ be a 3-manifold with torus boundary components $T_1$ and $T_2$. Let
$\phi \colon T_1 \to T_2$ be a homeomorphism, $M_\phi$ the manifold obtained
from $M$ by gluing $T_1$ to $T_2$ via the map $\phi$, and $T$ the image of
$T_1$ in $M_\phi$. We show that if $\phi$ is "sufficiently complicated" then
any incompressible or strongly irreducible surface in $M_\phi$ can be isotoped
to be disjoint from $T$. It follows that every Heegaard splitting of a
3-manifold admitting a "sufficiently complicated" JSJ decomposition is an
amalgamation of Heegaard splittings of the components of the JSJ decomposition.
