Abstract Bell’s theorem renders quantum correlations distinct from those of any local-realistic model. Although being stronger than classical correlations, quantum correlations are limited by the Tsirelson bound. This bound, however, applies for Hermitian, commutative operators corresponding to non-signaling observables in Alice’s and Bob’s spacelike-separated labs. As an attempt to explore theories beyond quantum mechanics and analyze the uniqueness of the latter, we examine in this work the extent of non-local correlations when relaxing these fundamental assumptions, which allows for theories with non-local signaling. We prove that, somewhat surprisingly, the Tsirelson bound in the Bell–Clauser–Horne–Shimony–Holt scenario, and similarly other related bounds on non-local correlations, remain effective as long as we maintain the Hilbert space structure of the theory. Furthermore, in the case of Hermitian observables we find novel relations between non-locality, local correlations, and signaling. We demonstrate that such non-local signaling theories are naturally simulated by quantum systems of parafermionic zero modes. We numerically study the derived bounds in parafermionic systems, confirming the bounds’ validity yet finding a drastic difference between correlations of ‘signaling’ and ‘non-signaling’ sets of observables. We also propose an experimental procedure for measuring the relevant correlations.