  The existence of global nonnegative martingale solutions to a stochastic
cross-diffusion system for an arbitrary but finite number of interacting
population species is shown. The random influence of the environment is modeled
by a multiplicative noise term. The diffusion matrix is generally neither
symmetric nor positive definite, but it possesses a quadratic entropy
structure. This structure allows us to work in a Hilbert space framework and to
apply a stochastic Galerkin method. The existence proof is based on energy-type
estimates, the tightness criterion of Brze\'zniak and co-workers, and
Jakubowski's generalization of the Skorokhod theorem. The nonnegativity is
proved by an extension of Stampacchia's truncation method due to Chekroun,
Park, and Temam.
