We consider Bayesian multiple testing of equality of two proportions, under a “control” group and a “treatment” group, across multiple conditions when these proportions are potentially correlated across conditions. A hierarchical prior distribution for the proportions that accounts for the correlation and allows for multiplicity adjustment is specified in two stages. In the first stage, the correlation across conditions is accounted for by the use of a conditional autoregressive type prior for the proportions in the control group, and in the second stage an objective prior for the proportions under the alternatives is specified, conditional on the proportions under the controls.  Simulation studies were carried out to assess the performance of the proposed approach, and to compare its frequentist characteristics with those of a Bayesian approach assuming independence and certain commonly used frequentist methods. We also illustrate the approach using a real data set. Overall, we found that the proposed Bayesian approach to have certain advantages over the other approaches considered, and that accounting for correlation when it exists improved performance.