A theoretical analysis is provided of the continuity of multiphase compositional space parameterization for thermal-compositional reservoir simulation. It is shown that the tie-simplex space changes continuously as a function of composition, pressure, and temperature, and this justifies the Compositional Space Adaptive Tabulation (CSAT) framework, in which a discrete number of tie-simplexes are constructed, tabulated, and reused in the course of a simulation. The CSAT is extended for thermal-compositional displacements for mixtures that can form an arbitrary number of phases. In particular, the construction is described of three-phase tie-simplex tables, and it is shown how the degeneration of multiphase regions can be accurately captured over wide ranges of temperature and pressure. Several challenging multiphase examples are used to demonstrate the accuracy and effectiveness of phase-state identification using tabulated tie-simplexes. (C) 2012 American Institute of Chemical Engineers AIChE J, 59: 1684-1701, 2013