Completely random measures (CRMs) have been well-studied as convenient priors for Bayesian nonparametrics (BNP). It is common to define a CRM prior on a countably infinite set of rates via a Poisson point process with rate measure on the non-negative real line. This prior is often paired with a count likelihood. We consider the case where the rate measure is instead over non-negative orthant, which can be interpreted as generating a vector of rates in D dimensions. For instance, these could represent the rates of genetic variants in D populations, and might be paired with a Bernoulli likelihood for a full generative model of genetic variants. We show that, surprisingly, the choice of a rate measure that factorizes across dimensions fails to satisfy natural BNP desiderata: roughly, that each sample is finite but that there are always more features to discover in each dimension. We propose an alternative construction that satisfies these desiderata while maintaining exponential conjugacy. We develop tools to characterize the behavior of the number of observed features as the sample size grows across dimensions. And we provide conditions that dictate realistic power-law growth.