Non-signalling conditions encode minimal requirements that any (quantum)
systems put into spatial arrangements must satisfy in order to be consistent
with special relativity. Recent works have argued that in scenarios involving
more that two parties, conditions compatible with relativistic causality do not
have to satisfy all possible non-signalling conditions but only a subset of
them. Here we show that correlations satisfying only this subset of constraints
have to satisfy highly non-local monogamy relations between the effects of
space-like separated random variables. These monogamy relations take the form
of new entropic inequalities between the various systems and we give a general
method to derive them. Using these monogamy relations we refute previous
suggestions for physical mechanisms that could lead to relativistically causal
correlations, demonstrating that such mechanisms would lead to superluminal
signalling.Comment: 14 pages, 4 figure