  We consider a 2+1 dimensional model of charged fermions coupled to a
$\mathbb{Z}_2$ gauge field, and study the confinement transition in this
regime. To elucidate the phase diagram of this model, we introduce a method to
handle the Gauss law constraint within sign problem free determinantal quantum
Monte Carlo, at any charge density. For generic charge densities,
$\mathbb{Z}_2$ gauge fluctuations mediate pairing and the ground state is a
gapped superfluid. Superfluidity also appears in the confined phase. This is
reminiscent of the BCS-BEC crossover, except that a true zero temperature
transition occurs here, with the maximum $T_c$ achieved near the transition. At
half-filling also one obtains a large Fermi surface which is gapped at zero
temperature. However, on increasing fermion hopping a $\pi$-flux phase is
spontaneously generated, with emergent Dirac fermions that are stable against
pairing. In contrast to a Fermi liquid of electrons, the change in Fermi
surface volumes of the $\mathbb{Z}_2$ fermions occurs without the breaking of
translation symmetry. Unexpectedly, the numerics indicate a single continuous
transition between the deconfined Dirac phase and the confined superfluid, in
contrast to the naive expectation of a split transition, where a gap to
fermions precedes confinement.
