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A positivity-preserving fractional algorithm is presented for solving the four-equation homogeneous relaxation model (HRM) with an arbitrary number of ideal gases and a liquid governed by the stiffened gas equation of state. The fractional algorithm consists of a time step of the hyperbolic five-equation model by Allaire et al. and an algebraic numerical thermal relaxation step at an infinite relaxation rate. Interpolation and flux limiters are proposed for the use of high-order Cartesian finite difference or finite volume schemes in a general form such that the positivity of the partial densities and squared sound speed, as well as the boundedness of the volume fractions and mass fractions, are preserved with the algorithm. A conservative solution update for the four-equation HRM is also guaranteed by the algorithm which is advantageous for certain applications such as those with phase transition. The accuracy and robustness of the algorithm with a high-order explicit finite difference weighted compact nonlinear scheme (WCNS) using the incremental-stencil weighted essentially non-oscillatory (WENO) interpolation, are demonstrated with various numerical tests.