<p>This paper presents a physical-constraint-preserving quasi-conservative discontinuous Galerkin (DG) method tailored for solving compressible two-phase flows governed by the six-equation with stiffened gas equations of state. We integrate the quasi-conservative DG scheme with the extended central-upwind numerical flux to effectively address the model. Additionally, we employ an affine-invariant weighted essentially non-oscillatory limiter to suppress numerical oscillation near discontinuities. To guarantee that the physical quantities remain within bounds and ensure numerical stability, we propose and analyze a bound-preserving limiting strategy for volume fraction and a positivity-preserving limiting procedure for the density of each phase and internal energy. Numerical experiments conducted in one- and two-dimensional spaces demonstrate the accuracy, robustness, and various properties such as equilibrium-, bound-, and positivity-preserving of the proposed scheme.</p>