Building on our previously introduced multicell Monte Carlo (MC)^{2} method for modeling phase coexistence, this paper provides important improvements for efficient determination of phase equilibria in solids. The (MC)^{2} method uses multiple cells, representing possible phases. Mass transfer between cells is modeled virtually by solving the mass balance equation after the composition of each cell is changed arbitrarily. However, searching for the minimum free energy during this process poses a practical problem. The solution to the mass balance equation is not unique away from equilibrium, and consequently the algorithm is in risk of getting trapped in nonequilibrium solutions. Therefore, a proper stopping condition for (MC)^{2} is currently lacking. In this work, we introduce a consistency check via a predictor-corrector algorithm to penalize solutions that do not satisfy a necessary condition for equivalence of chemical potentials and steer the system toward finding equilibrium. The most general acceptance criteria for (MC)^{2} is derived starting from the isothermal-isobaric Gibbs ensemble for mixtures. Using this ensemble, translational MC moves are added to include vibrational excitations as well as volume MC moves to ensure the condition of constant pressure and temperature entirely with a MC approach, without relying on any other method for relaxation of these degrees of freedom. As a proof of concept the method is applied to two binary alloys with miscibility gaps and a model quaternary alloy, using classical interatomic potentials.