  Crystallization is perhaps the most familiar example of a symmetry-breaking
transition. In equilibrium, thermodynamic arguments result in a powerful and
convenient set of criteria for determining the coexistence curves associated
with these transitions. In recent years, nonequilibrium symmetry-breaking
transitions have been routinely observed in a variety of natural and synthetic
systems. The breaking of detailed balance, and the resulting absence of
Boltzmann statistics, motivates the need for a symmetry-breaking coexistence
theory that is independent of the underlying distribution of microstates. Here,
we develop such a theory, relying only on mechanics, balance laws, and system
symmetries. In doing so, we develop a generalized Gibbs-Duhem relation that
results in nonequilibrium coexistence criteria solely in terms of bulk
equations of state. We apply our framework to active crystallization,
developing a complete description of the phase diagram of active Brownian hard
spheres. Our predicted phase diagram quantitatively recapitulates the
solid-fluid coexistence curve as well as other key features of active phase
behavior, such as the liquid-gas coexistence binodal and solid-liquid-gas
triple point. It is our hope that our findings offer a concrete path forward
towards the development of a general theory for nonequilibrium coexistence.
