We consider cross-diffusion systems describing evolution of two species u and v moving according to the Darcy’s law with the power-law pressure depending on the sum u+v. One of the most challenging questions in the field is the construction of solutions to the problem in the presence of additional advection fields, without imposing any artificial structure (for instance, preventing the species from mixing). Although advection arises naturally in these models, it breaks the symmetry of the system and prevents application of techniques developed in recent years. We solve the problem in one space dimension in a unified way for all pressure exponents (fast-diffusion and porous medium) and arbitrary initial data (segregated, mixed, or partially mixed). In the porous medium regime, our work provides the first existence result without structural assumptions. We construct the solutions as a limit of a vanishing viscosity approximation. The main challenge is to identify the limit of the product and the key new insight is that possible oscillations of its components are correlated, which simplifies the analysis of the associated Young measures in the compensated compactness argument and yields strong convergence of both quantities. Quite surprisingly, the argument relies on only three entropy/entropy-flux pairs.