A global algorithm for implementing the Gibbs tangent plane stability test in multicomponent, multiphase liquid mixtures is described. The algorithm is self-starting and significantly improves the reliability and robustness of multiphase equilibrium calculations. The main improvement is a result of a new approach for locating the stationary points of the tangent plane distance function. This relies on the geometric properties of the tangent plane distance function but is independent of the specific fluid model. For ternary mixtures, it is shown that stationary points in the tangent plane distance function satisfy a global topological constraint N-max + N-min - N-sad = 1 where N-max is the number of maxima, N-min is the number of minima, and N-sad is the number of saddles. This provides a global consistency check on the numerical results that is independent of the specific model or parameters used to describe the mixture. The method has been implemented and tested on a variety of problems with three and four components and up to three liquid phases. Solutions can be found with sufficient speed for interactive design and modeling. In all of the examples, solutions for the entire phase diagram were found without fail, and certain problems that are difficult or impossible to solve using other methods were modeled successfully.