Recent studies suggest that selection can allow coexistence in situations where ecological dynamics lead to competitive exclusion, provided that there is a trade-off between traits optimal for interacting with conspecifics and traits optimal for interacting with heterospecifics. Despite compelling empirical evidence, there is no general framework for elucidating how and when selection will allow coexistence in natural communities. Here we develop such a framework for a mechanism that we term "neighbor-dependent selection." We show that this mechanism can both augment coexistence when ecological conditions allow for niche partitioning and enable coexistence when ecological conditions lead to competitive exclusion. The novel insight is that when ecological conditions lead to exclusion, neighbor-dependent selection can allow coexistence via cycles driven by an intransitive loop; selection causes one species to be a superior interspecific competitor when it is rare and an inferior interspecific competitor when it is abundant. Our framework predicts the conditions under which selection can enable coexistence, as opposed to merely augmenting it, and elucidates the effects of heritability on the eco-evolutionary feedbacks that drive coexistence. Given increasing evidence that evolution operates on ecological timescales, our approach provides one means for evaluating the role of selection and trait evolution in species coexistence.