  The existence of global-in-time bounded martingale solutions to a general
class of cross-diffusion systems with multiplicative Stratonovich noise is
proved. The equations describe multicomponent systems from physics or biology
with volume-filling effects and possess a formal gradient-flow or entropy
structure. This structure allows for the derivation of almost surely positive
lower and upper bounds for the stochastic processes. The existence result holds
under some assumptions on the interplay between the entropy density and the
multiplicative noise terms. The proof is based on a stochastic Galerkin method,
a Wong--Zakai type approximation of the Wiener process, the
boundedness-by-entropy method, and the tightness criterion of Brze\'{z}niak and
coworkers. Three-species Maxwell--Stefan systems and $n$-species biofilm models
are examples that satisfy the general assumptions.
