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Predicting food web structure in future climates is a pressing goal of ecology. These predictions may be impossible without a solid understanding of the factors that structure current food webs. The most fundamental aspect of food web structure—the relationship between the number of links and species—is still poorly understood. Some species interactions may be physically or physiologically ‘forbidden'—like consumption by non-consumer species—with possible consequences for food web structure. We show that accounting for these ‘forbidden interactions' constrains the feasible link-species space, in tight agreement with empirical data. Rather than following one particular scaling relationship, food webs are distributed throughout this space according to shared biotic and abiotic features. Our study provides new insights into the long-standing question of which factors determine this fundamental aspect of food web structure.