Realized covariance measures (RCs) are an essential input to assess the risks involved in different investment allocations and it is thus useful to model and forecast them. Thus, a realistic distributional assumption is essential. We compare all probability distributions hitherto applied to time series of RCs in the literature. We derive them in an intuitive and unified framework based on their stochastic representations in terms of random lower and upper triangular (Barlett) matrices. Furthermore, we derive a novel family of probability distributions, which has a property called tail homogeneity That is, in times of crisis periods, i.e.~large RCs, this family assumes high dependence between the individual entries in the RCs. Finally, we show rigorously, how the considered distributions are related to each other. Empirically, we confirm in an in-sample fit experiment previous results that at-tailed distributions outperform others and show that the novel distribution family achieves a very good fit. Out-of-sample forecasting comparisons further corroborate the excellent performance of the novel distribution family.