  This paper studies the structural implications of constant vorticity for
steady three-dimensional internal water waves. It is known that in many
physical regimes, water waves beneath vacuum that have constant vorticity are
necessarily two dimensional. The situation is more subtle for internal waves
that traveling along the interface between two immiscible fluids. When the
layers have the same density, there is a large class of explicit steady waves
with constant vorticity that are three-dimensional in that the velocity field
and pressure depend on one horizontal variable while the interface is an
arbitrary function of the other. We prove the following rigidity result: every
three-dimensional traveling internal wave with bounded velocity for which the
vorticities in the upper and lower layers are nonzero, constant, and parallel
must belong to this family. If the densities in each layer are distinct, then
in fact the flow is fully two dimensional.
