Lattice gauge theories are used to describe a wide range of phenomena from quark confinement to quantum materials. At finite fermion density, gauge theories are notoriously hard to analyse due to the fermion sign problem. Here, we investigate the Ising gauge theory in 2 + 1 dimensions, a problem of great interest in condensed matter, and show that it is free of the sign problem at arbitrary fermion density. At generic filling, we find that gauge fluctuations mediate pairing, leading to a transition between a deconfined BCS state and a confined BEC. At half-filling, a €-flux phase is generated spontaneously with emergent Dirac fermions. The deconfined Dirac phase, with a vanishing Fermi surface volume, is a non-trivial example of violation of Luttinger's theorem due to fractionalization. At strong coupling, we find a single continuous transition between the deconfined Dirac phase and the confined BEC, in contrast to the expected split transition.