In clinical trials of infectious disease prevention methods, community-level randomization represents a natural study design choice: it allows researchers to evaluate interventions that are defined at a supra-individual level, while reducing the possibility of treatment contamination between arms. However, the outcomes from these studies are necessarily cluster-correlated; in the event that they are only recorded at intermittent study visits, then they are also interval-censored. Here we present an expectation maximization algorithm that permits semiparametric estimation of stratified proportional hazards models to data that are both clustered and interval censored. We also consider its extensions to settings in which covariates may vary over time due to, for example, a study-mandated treatment switch or a change in national treatment guidelines. Finally, we demonstrate the performance of our method using data from the Botswana Combination Prevention Project, a large cluster-randomized trial of combination HIV prevention strategies.