We study a parabolic population model in the full space and prove the global-in-time existence of a weak solution. This model consists of two strongly coupled diffusion equations describing the population densities of two competing species. The system features intrinsic growth, interspecies and intraspecies competition of the species, as well as diffusion, cross-diffusion and self-diffusion, and drift terms related to varying environment quality. The cross-diffusion terms can be large, making the system non-parabolic for large initial data. For the first time to our knowledge, solutions in unbounded domains are studied. The method of our proof is a combination of a time semi-discretization, a transformation of the system guaranteeing the positivity of the solution, a special entropy symmetrizing the system and compactness arguments.