  We study equilibrium properties of binary lattice-gases comprising $A$ and
$B$ particles, which undergo continuous exchanges with their respective
reservoirs, maintained at chemical potentials $\mu_A = \mu_B = \mu$. The
particles interact via on-site hard-core exclusion and also between the
nearest-neighbours: there are a soft repulsion for $AB$ pairs and interactions
of arbitrary strength $J$, positive or negative, for $AA$ and $BB$ pairs. For
tree-like Bethe and Husimi lattices, we determine the full phase diagram of
such a ternary mixture of particles and voids. We show that for $J$ being above
a lattice-dependent threshold value, the critical behaviour is similar: the
system undergoes a transition at $\mu = \mu_c$ from a phase with equal mean
densities of species into a phase with a spontaneously broken symmetry, in
which the mean densities are no longer equal. Depending on the value of $J$,
this transition can be either continuous or of the first order. For
sufficiently big negative $J$, the behaviour on the two lattices becomes
markedly different: while for the Bethe lattice there exists a continuous
transition into a phase with an alternating order followed by a continuous
re-entrant transition into a disordered phase, an alternating order phase is
absent on the Husimi lattice due to strong frustration effects.
