In a single-component superfluid under rotation a broken symmetry in the order parameter space results in a broken translational symmetry in real space: a vortex lattice. If translational symmetry is restored, the phase of the order parameter disorders and thus the broken symmetry in the order parameter space is also restored. We show that for Bose-Einstein condensate mixtures in optical lattices with negative dissipationless drag, a new situation arises. This state is a modulated vortex liquid which breaks translational symmetry in the direction transverse to the rotation vector.