Models of sexually transmitted infections have become a fixture of mathematical epidemiology. A common attribute of all these models is treating reproduction and mating, and hence pathogen transmission, as uncoupled events. This is fine for humans, for example, where only a tiny fraction of sexual intercourses ends up with having a baby. But it can be a deficiency for animals in which mating and giving birth are tightly coupled, and mating thus mediates both reproduction and pathogen transmission. Here, we model dynamics of sterilizing, sexually transmitted infections in such animals, assuming structural consistency between the processes of reproduction and pathogen transmission. We show that highly sterilizing, sexually transmitted pathogens trigger bistability in the host population. In particular, the host population can end up in two extreme alternative states, disease-free persistence and pathogen-driven extinction, depending on its initial state. Given that sterilizing, sexually transmitted infections that affect animals are abundant, our results might implicate an effective pest control tactic that consists of releasing the corresponding pathogens, possibly after genetically enhancing their sterilization power.