In order to meet regulatory approval, pharmaceutical companies often must demonstrate that new vaccines reduce the total risk of a post-infection outcome like transmission, symptomatic disease, severe illness, or death in randomized, placebo-controlled trials. Given that infection is a necessary precondition for a post-infection outcome, one can use principal stratification to partition the total causal effect of vaccination into two causal effects: vaccine efficacy against infection, and the principal effect of vaccine efficacy against a post-infection outcome in the patients that would be infected under both placebo and vaccination. Despite the importance of such principal effects to policymakers, these estimands are generally unidentifiable, even under strong assumptions that are rarely satisfied in real-world trials. We develop a novel method to nonparametrically point identify these principal effects while eliminating the monotonicity assumption and allowing for measurement error. Furthermore, our results allow for multiple treatments, and are general enough to be applicable outside of vaccine efficacy. Our method relies on the fact that many vaccine trials are run at geographically disparate health centers, and measure biologically-relevant categorical pretreatment covariates. We show that our method can be applied to a variety of clinical trial settings where vaccine efficacy against infection and a post-infection outcome can be jointly inferred. This can yield new insights from existing vaccine efficacy trial data and will aid researchers in designing new multi-arm clinical trials.