  We investigate the conditions under which a set of multipartite nonlocal
correlations can describe the distributions achievable by distant parties
conducting experiments in a consistent universe. Several questions are posed,
such as: are all such sets "nested", i.e., contained into one another? Are they
discrete or do they form a continuum? How many of them are supraquantum? Are
there non-trivial polytopes among them? We answer some of these questions or
relate them with established conjectures in complexity theory by introducing a
"zoo" of physically consistent sets which can be characterized efficiently via
either linear or semidefinite programming. As a bonus, we use the zoo to
derive, for the first time, concrete impossibility results in nonlocality
distillation.
