The existence of global nonnegative martingale solutions to cross-diffusion systems of Shigesada-Kawasaki-Teramoto type with multiplicative noise is proven. The model describes the stochastic segregation dynamics of an arbitrary number of population species in a bounded domain with no-flux boundary conditions. The diffusion matrix is generally neither symmetric nor positive semidefinite, which excludes standard methods for evolution equations. Instead, the existence proof is based on the entropy structure of the model, a novel regularization of the entropy variable, higher-order moment estimates, and fractional time regularity. The regularization technique is generic and is applied to the population system with self-diffusion in any space dimension and without self-diffusion in two space dimensions.