  We present a class of states with both topological and conventional Landau
order that arise out of strongly interacting spinless fermions in fractionally
filled and topologically non-trivial bands with Chern number $C=\pm 1$. These
quantum states show the features of fractional Chern insulators, such as
fractional Hall conductivity and interchange of ground-state levels upon
insertion of a magnetic flux. In addition, they exhibit charge order and a
related additional trivial ground-state degeneracy. Band mixing and geometric
frustration of the charge pattern place these lattice states markedly beyond a
single-band description.
