A numerical method on unstructured Eulerian grids was proposed recently by the authors for two-phase compressible flows with ideal gases [Adv. Appl. Math. Mech., 8(2), 2016, pp. 187--212]. We extend the method therein to the case with fluids described by the Mie-Gr\"uneisen equation of state, which is widely used in practical applications, and propose a robust approximate Riemann solver in an iterative manner to obtain the interface velocity and fluxes. The accuracy of the approximate Riemann solver is analyzed. Numerical examples are presented for Riemann problems, air blast and underwater explosion applications. It is shown that our method leads to reliable and high resolution for shocks and phase interfaces.