We develop a Godunov-type scheme for a non-conservative, unconditional hyperbolic multiphase model. It involves a set of seven partial differential equations and has the ability to solve interface problems between pure materials as well as compressible multiphase mixtures with two velocities and non-equilibrium thermodynamics (two pressures, two temperatures, two densities, etc.). Its numerical resolution poses several difficulties. The model possesses a large number of acoustic and convective waves (seven waves) and it is not easy to upwind all these waves accurately and simply. Also, the system is non-conservative, and the numerical approximations of the corresponding terms need to be provided. In this paper we focus on a method, based on a characteristic decomposition which solves these problems in a simple way and with good accuracy. The robustness, accuracy and versatility of the method is clearly demonstrated on several test problems with exact solutions.