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We study equilibrium properties of binary lattice-gases comprising $A$ and$B$ particles, which undergo continuous exchanges with their respectivereservoirs, maintained at chemical potentials $\mu_A = \mu_B = \mu$. Theparticles interact via on-site hard-core exclusion and also between thenearest-neighbours: there are a soft repulsion for $AB$ pairs and interactionsof arbitrary strength $J$, positive or negative, for $AA$ and $BB$ pairs. Fortree-like Bethe and Husimi lattices, we determine the full phase diagram ofsuch a ternary mixture of particles and voids. We show that for $J$ being abovea lattice-dependent threshold value, the critical behaviour is similar: thesystem undergoes a transition at $\mu = \mu_c$ from a phase with equal meandensities of species into a phase with a spontaneously broken symmetry, inwhich the mean densities are no longer equal. Depending on the value of $J$,this transition can be either continuous or of the first order. Forsufficiently big negative $J$, the behaviour on the two lattices becomesmarkedly different: while for the Bethe lattice there exists a continuoustransition into a phase with an alternating order followed by a continuousre-entrant transition into a disordered phase, an alternating order phase isabsent on the Husimi lattice due to strong frustration effects.

Binary lattice-gases of particles with soft exclusion: Exact phase diagrams for tree-like lattices