A new algorithm to determine phase equilibria at the global minimum of the Gibbs free energy is presented. Using the Newton homotopy-continuation method, when checking for phase stability, multiple stationary points of the tangent-plane-distance function (TPDF) are determined. Subsets of these are selected as estimates for the minimization of the Gibbs free energy. The algorithm, implemented in the HOMPEQ computer program, was tested for nine non-ideal mixtures. The homotopy paths, when initialized with nearly pure species, reliably located all of the stationary points of the TPDF. This was accomplished for mixtures that exhibit multiple immiscibility gaps. The combined algorithm located the global minimum for all of the systems involving symmetric and asymmetric models, with a final stability analysis implemented to ensure global minimization. In comparison with Michelsen’s programs, HOMPEQ is slower and provides comparable reliability, but is initialized with nearly pure phases, a much less elaborate initialization strategy.