We have studied by quantum Monte Carlo simulations the low temperature phase diagram of a mixture of isotopic, hard core bosons, described by the t-Jz-J⊥ model, with J⊥=αJz. The coexistence of superfluid hole-rich and insulating, antiferromagnetically ordered hole-free phases is observed at sufficiently low hole density for any α<1. A two-component checkerboard supersolid phase is not observed. The experimental relevance and possible broader implications of these findings are discussed.