The authors introduce a new model for the dynamics of fat-tailed (realized) covariance-matrix-valued time-series using the F-Riesz distribution. The model allows for heterogeneous tail behavior across the coordinates of the covariance matrix via two vector-valued degrees of freedom parameters, thus generalizing the familiar Wishart and matrix-F distributions. We show that the filter implied by the new model is invertible and that a two-step targeted maximum likelihood estimator is consistent. Applying the new F-Riesz model to U.S. stocks, both tail heterogeneity and tail fatness turn out to be empirically relevant: they produce significant in-sample and out-of-sample likelihood increases, ex-post portfolio standard deviations that are in the global minimum variance model confidence set, and economic differences that are either in favor of the new model or competitive with a range of benchmark models.