In recent years, the use of information principles to understand quantum correlations has been very successful. Unfortunately, all principles considered so far have a bipartite formulation, but intrinsically multipartite principles, yet to be discovered, are necessary for reproducing quantum correlations. Here we introduce local orthogonality, an intrinsically multipartite principle stating that events involving different outcomes of the same local measurement must be exclusive or orthogonal. We prove that it is equivalent to no-signalling in the bipartite scenario but more restrictive for more than two parties. By exploiting this non-equivalence, it is then demonstrated that some bipartite supra-quantum correlations do violate the local orthogonality when distributed among several parties. Finally, we show how its multipartite character allows revealing the non-quantumness of correlations for which any bipartite principle fails. We believe that local orthogonality is a crucial ingredient for understanding no-signalling and quantum correlations.