We introduce a Potts model with quenched, frustrated disorder, which enjoys a gauge symmetry that forbids spontaneous magnetization and allows the glassy phase to extend from Tc down to T50. We study numerically the four-dimensional model with q54 states. We show the existence of a glassy phase, and we characterize it by studying the probability distributions of an order parameter, the binder cumulant, and the divergence of the overlap susceptibility. We show that the dynamical behavior of the system is characterized by aging. @S0163-1829~99!10013-4#