  We present an $\textit{ab initio}$, numerically exact study of attractive
fermions in square lattices with Rashba spin-orbit coupling. The ground state
of this system is a supersolid, with co-existing charge and superfluid order.
The superfluid is composed of both singlet and triplet pairs induced by
spin-orbit coupling. We perform large-scale calculations using auxiliary-field
quantum Monte Carlo to provide the first full, quantitative description of the
charge, spin, and pairing properties of the system. In addition to
characterizing the exotic physics, our results will serve as essential
high-accuracy benchmarks for the intense theoretical and especially
experimental efforts in ultracold atoms to realize and understand an expanding
variety of quantum Hall and topological superconductor systems.
