  We present the theoretical mean-field zero-temperature phase diagram of a
Bose-Einstein condensate (BEC) with dipolar interactions loaded into an optical
lattice with a staggered flux. Apart from uniform superfluid, checkerboard
supersolid and striped supersolid phases, we identify several supersolid phases
with staggered vortices, which can be seen as combinations of supersolid phases
found in earlier work on dipolar BECs and a staggered-vortex phase found for
bosons in optical lattices with staggered flux. By allowing for different
phases and densities on each of the four sites of the elementary plaquette,
more complex phase patterns are found.
