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. https://www.shopmillensiumers.shop/product-category/terpenes/