  We study the phase diagram of the Bose-Hubbard model on the kagome lattice
with a broken sublattice symmetry. Such a superlattice structure can naturally
be created and tuned by changing the potential offset of one sublattice in the
optical generation of the frustrated lattice. The superstructure gives rise to
a rich quantum phase diagram, which is analyzed by combining Quantum Monte
Carlo simulations with the Generalized Effective Potential Landau Theory. Mott
phases with non-integer filling and a characteristic order along stripes are
found, which show a transition to a superfluid phase with an anisotropic
superfluid density. Surprisingly, the direction of the superfluid anisotropy is
changing between different symmetry directions as a function of the particle
number or the hopping strength. Finally, we discuss characteristic signatures
of anisotropic phases in time-of-flight absorption measurements.
