Abstract Predators often consume multiple prey and by mutually subsidizing a shared predator, the prey may reciprocally harm each other. When predation levels are high, this apparent competition can culminate in a prey species being displaced. Coupling quantitative genetics and Lotka–Volterra models, we study how predator evolution alters this and other ecological outcomes. These models account for a trade‐off between the predator's attack rates on two prey species. We provide a mathematical characterization of a strong form of persistence–permanence—for which there is a global attractor bounded away from extinction. When the evolutionary dynamics occur at a sufficiently slower time scale than the ecological dynamics, we also characterize attractors and their basins of attraction using singular perturbation theory and a graphical approach to the eco‐evolutionary dynamics. Our results show that eco‐evolutionary feedbacks can mediate permanence at intermediate trade‐offs in the attack rates. However, at strong trade‐offs, permanence is lost. Despite this loss of permanence, there can be attractors supporting coexistence. These attractors, however, may coincide with attractors at which the predator is excluded. Our results highlight that eco‐evolutionary feedbacks can alter community structure by mediating coexistence or leading to trait‐dependent alternative stable states.