Motivated by the mathematical modeling of tumor invasion in healthy tissues, we propose a generalized compressible diphasic Navier-Stokes Cahn-Hilliard model that we name G-NSCH. We assume that the two phases of the fluid represent two different populations of cells: cancer cells and healthy tissue. We include in our model possible friction and proliferation effects. The model aims to be as general as possible to study the possible mechanical effects playing a role in invasive growth of a tumor. In the present work, we focus on the analysis and numerical simulation of the G-NSCH model. Our G-NSCH system is derived rigorously and satisfies basic mechanics of fluids and thermodynamics of particles. Under simplifying assumptions, we prove the existence of global weak solutions. We also propose a structure preserving numerical scheme based on the scalar auxiliary variable method to simulate our system and present some numerical simulations validating the properties of the numerical scheme and illustrating the solutions of the G-NSCH model.