Individual social identities indicate group affiliations and are typically associated with group-typical preferences, signals that indicate group membership, and the propensity to condition actions on the social signals of others, resulting in group-differentiated interaction norms. Past work modeling identity signaling and coordination has typically assumed that individuals belong to one of a discrete set of groups. Yet individuals can simultaneously belong to multiple groups, which may be nested within larger groupings. Here, we introduce the generalized Bach or Stravinsky game, a coordination game with ordered preferences, which allows us to construct a model that captures the overlapping and hierarchical nature of social identity. Our model unifies several prior results into a single framework, including results related to coordination, minority disadvantage, and cross-cultural competence. Our model also allows agents to express complex social identities through multidimensional signaling, which we use to explore a variety of complex group structures. Our consideration of intersectional identities exposes flaws in naive measures of group structure, illustrating how empirical studies may overlook some social identities if they do not consider the behaviors that those identities function to afford.