In the present paper, a fully coupled numerical model is developed for the hydromechanical analysis of deforming, progressively fracturing porous media interacting with the flow of two immiscible, compressible wetting and non-wetting pore fluids. The governing equations involving the coupled two-phase fluid flow and deformation processes in partially saturated porous media containing cohesive cracks are derived within the framework of the generalized Biot theory. The displacement of the solid phase, the pressure of the wetting phase and the capillary pressure are taken as the primary unknowns of the three-phase formulation. A softening cohesive law is employed to describe the nonlinear behavior of the material in the fracture process zone. In order to account for the flux of the two fluid phases through the fracture faces, the mass balance equation for each flowing fluid inside the fully damaged zone and the cohesive zone is averaged over its cross section. The resulting equations provide mass couplings to the standard equations of the multiphase system. The effect of cracking and therefore change of porosity on the permeability of the damaged zone is also taken into account. To arrive at the discrete equations, the extended finite element method (XFEM) is utilized to discretize the weak form of the balance equations of mass and linear momentum in spatial domain along with the Generalized Newmark scheme for time domain discretization. By exploiting the partition of unity property of finite element shape functions, the evolving cohesive crack is simulated independently of the underlying finite element mesh and without continuous remeshing of the domain as the crack grows by adding enriched degrees of freedom to nodes whose support is bisected by the crack. For the numerical solution, the unconditionally stable direct time-stepping procedure is applied to solve the resulting system of strongly coupled non-linear algebraic equations using a Newton-Raphson iterative procedure. Finally, numerical simulations are presented to demonstrate the capability of the proposed method and the significant influence of the hydro-mechanical coupling between the continuum porous medium and the discontinuity on the results.