  We investigate a class of two-dimensional two-band microscopic models in
which the inter-band repulsive interactions play the dominant role. We first
demonstrate three different schemes of constraining the ratios between the
three types of inter-band interactions -- density-density, spin exchange, and
pair-hopping -- that render the model free of the fermionic sign-problem for
any filling and, consequently, amenable to efficient Quantum Monte Carlo
simulations. We then study the behavior of these sign-problem-free models in
the strong-coupling regime. In the cases where spin-rotational invariance is
preserved or lowered to a planar symmetry, the strong-coupling ground state is
a quantum paramagnet. However, in the case where there is only a residual Ising
symmetry, the strong-coupling expansion maps onto the transverse-field
$J_1$-$J_2$ Ising model, whose pseudospins are associated with local inter-band
magnetic order. We show that by varying the band structure parameters within a
reasonable range of values, a variety of ground states and quantum critical
points can be accessed in the strong-coupling regime, some of which are not
realized in the weak-coupling regime. We compare these results with the case of
the single-band Hubbard model, where only intra-band repulsion is present, and
whose strong-coupling behavior is captured by a simple Heisenberg model.
