We present approaches to generating synthetic microdata for multivariate data that take on non-negative integer values, such as magnitude data in economic surveys. The basic idea is to estimate a mixture of Poisson distributions to describe the multivariate distribution, and release draws from the posterior predictive distribution of the model. We develop approaches that guarantee the synthetic data sum to marginal totals computed from the original data, as well approaches that do not enforce this equality. For both cases, we present methods for assessing disclosure risks inherent in releasing synthetic magnitude microdata. We illustrate the methodology using economic data from a survey of manufacturing establishments.