Multivariate interval-censored data often arise in cluster-randomized trials where the outcome of interest is an asymptomatic event. For example, in cluster-randomized HIV prevention studies, the presence of an infection is determined through periodic serological testing, producing correlated and interval-censored observations. One natural choice for modeling these data is the mixed-effects proportional hazards model. However, most current algorithms for fitting these models to interval-censored data require that cluster sizes are small relative to the number of clusters---an assumption that often fails to hold in practice. Here we present a stochastic expectation maximization algorithm that permits semiparametric estimation of mixed-effects proportional hazards models with interval-censored data, even in settings where cluster membership is large. We also introduce a perturbation-resampling scheme to estimate the covariance matrix of the resulting estimators. Finally, we demonstrate the performance of our method using data modeled on the Botswana Combination Prevention Project, a large cluster-randomized trial of combination HIV prevention.