  We explore theoretically the nonequilibrium photonic phases of an array of
coupled cavities in presence of incoherent driving and dissipation. In
particular, we consider a Hubbard model system where each site is a Kerr
nonlinear resonator coupled to a two-level emitter, which is pumped
incoherently. Within a Gutzwiller mean-field approach, we determine the
steady-state phase diagram of such a system. We find that, at a critical value
of the inter-cavity photon hopping rate, a second-order nonequilibrium phase
transition associated with the spontaneous breaking of the $U(1)$ symmetry
occurs. The transition from an incompressible Mott-like photon fluid to a
coherent delocalized phase is driven by commensurability effects and not by the
competition between photon hopping and optical nonlinearity. The essence of the
mean-field predictions is corroborated by finite-size simulations obtained with
matrix product operators and corner-space renormalization methods.
