The existence of a global martingale solution to a cross-diffusion system with multiplicative Wiener noise in a bounded domain with no-flux boundary conditions is shown. The model describes the dynamics of population densities of different species due to segregation cross-diffusion effects. The diffusion matrix is generally neither symmetric nor positive semidefinite. This difficulty is overcome by exploiting the Rao entropy structure. The existence proof uses a stochastic Galerkin method, uniform estimates from the Rao entropy inequality, and the Skorokhod--Jakubowski theorem. Furthermore, an exponential equilibration result is proved for sufficiently small Lipschitz constants of the noise by using the relative Rao entropy. Numerical tests illustrate the behavior of solutions in one space dimension for two and three population species.