When conducting a meta-analysis involving prevalence data for an outcome with several subtypes, each of them is typically analyzed separately using a univariate meta-analysis model. Recently, multivariate meta-analysis models have been shown to correspond to a decrease in bias and variance for multiple correlated outcomes compared with univariate meta-analysis, when some studies only report a subset of the outcomes. In this article, we propose a novel Bayesian multivariate random effects model to account for the natural constraint that the prevalence of any given subtype cannot be larger than that of the overall prevalence. Extensive simulation studies show that this new model can reduce bias and variance when estimating subtype prevalences in the presence of missing data, compared with standard univariate and multivariate random effects models. The data from a rapid review on occupation and lower urinary tract symptoms by the Prevention of Lower Urinary Tract Symptoms Research Consortium are analyzed as a case study to estimate the prevalence of urinary incontinence and several incontinence subtypes among women in suspected high risk work environments.