  There have been a number of attempts to derive the set of quantum non-local
correlations from reasonable physical principles. Here we introduce
$\tilde{Q}$, a set of multipartite supra-quantum correlations that has appeared
under different names in fields as diverse as graph theory, quantum gravity and
quantum information science. We argue that $\tilde{Q}$ may correspond to the
set of correlations of a reasonable physical theory, in which case the research
program to reconstruct quantum theory from device-independent principles is met
with strong obstacles. In support of this conjecture, we prove that $\tilde{Q}$
is closed under classical operations and satisfies the physical principles of
Non-Trivial Communication Complexity, No Advantage for Nonlocal Computation,
Macroscopic Locality and Local Orthogonality. We also review numerical evidence
that almost quantum correlations satisfy Information Causality.
