  We introduce a family of lattice-gas models of flocking, whose
thermodynamically consistent dynamics admits a proper equilibrium limit at
vanishing self-propulsion. These models are amenable to an exact
coarse-graining which allows us to study their hydrodynamic behavior
analytically. We show that the equilibrium limit here belongs to the
universality class of Model C, and that it generically exhibits tricritical
behavior. Self-propulsion has a non-perturbative effect on the phase diagram,
yielding novel phase behaviors depending on the type of aligning interactions.
For aligning interactions that increase monotonically with the density, the
tricritical point diverges to infinite density reproducing the standard
scenario of a discontinuous flocking transition accompanied by traveling bands.
In contrast, for models where the aligning interaction is non-monotonic in
density, the system can exhibit either (the nonequilibrium counterpart of) an
azeotropic point, associated with a continuous flocking transition, or a state
with counterpropagating bands.
