Natural communities are well known to be maintained by many complex processes. Despite this, the practical aspects of studying them often require some simplification, such as the widespread assumption that direct, additive competition captures the important details about how interactions between species impact community diversity. More complex non-additive 'higher-order' interactions are assumed to be negligible or absent. Notably, these assumptions are poorly supported and have major consequences for the accuracy with which patterns of natural diversity are modelled and explained. We present a mathematically simple framework for incorporating biologically meaningful complexity into models of diversity by including non-additive higher-order interactions. We further provide empirical evidence that such higher-order interactions strongly influence species' performance in natural plant communities, with variation in seed production (as a proxy for per capita fitness) explained dramatically better when at least some higher-order interactions are considered. Our study lays the groundwork for a long-overdue shift in how species interactions are used to study the diversity of natural communities.