  A striking property of a single-component superfluid under rotation, is that
a broken symmetry in the order parameter results in a broken translational
symmetry, a vortex lattice. If translational symmetry is restored, the phase of
the order parameter disorders and the broken symmetry in the order parameter is
restored. We show that for Bose-Condensate mixtures on optical lattices (which
may possess a negative dissipationless intercomponent drag), a new situation
arises. A phase disordered nonsuperfluid component can break translational
symmetry in response to rotation due to interaction with a superfluid
component. This state is a modulated vortex liquid which breaks translational
symmetry in the direction transverse to the rotation vector.
