We model liquid-gas mixtures by a variant of the hyperbolic single-velocity six-equation two-phase compressible flow model with stiff pressure relaxation of Saurel--Petipas--Berry. One relevant feature of these liquid-gas flows is the large and rapid variation of the Mach number, since the speed of sound may range from very low values in the two-phase mixture to very large values in the liquid medium. Because of this, when classical upwind finite volume discretization of the compressible two-phase flow model are used, suitable strategies are needed to overcome the well known difficulty of loss of accuracy encountered by compressible flow solvers at low Mach regimes. To address this issue we study in this work extensions of the Roe-Turkel preconditioned method of Guillard--Viozat for the single-phase Euler equations to the considered two-phase flow model. First, taking advantage of our novel phasic-total-energy-based formulation of the six-equation model, we are able to derive a Roe matrix for the homogeneous model system that ensures conservation of phasic masses, mixture momentum and mixture total energy. Through a suitable choice of the primitive variables of the two-phase system to which the classical Turkel's preconditioner is applied, we then derive an expression of the preconditioned numerical dissipation tensor that naturally extends the form obtained for the single-phase case. In particular, only the acoustic characteristic fields are altered by preconditioning at low Mach number, while interface waves are preserved unchanged. The stiff mechanical relaxation source term of the two-phase model is treated via a fractional step algorithm and a pressure relaxation procedure that ensures consistency of the equilibrium pressure with the correct mixture equation of state. We present numerical results for a two-dimensional liquid-gas flow channel test that show the agreement of the solution of the relaxed two-phase model computed by the proposed Roe-Turkel scheme with the expected asymptotic behavior of the continuous relaxed model, corresponding to the limit for vanishing Mach number of the pressure-equilibrium five-equation model of Kapila et al. In particular, we show that in the low Mach number limit pressure fluctuations correctly scale with the square of the reference Mach number, in agreement with the theoretical results presented in the literature. (This work is joint with Keh-Ming Shyue, NTU)