  We study the set of almost quantum correlations and their refinements in the
simplest tripartite Bell scenario where each party is allowed to perform two
dichotomic measurements. In contrast to its bipartite counterpart, we find that
there already exist facet Bell inequalities that witness almost quantum
correlations beyond quantum theory in this simplest tripartite Bell scenario.
Furthermore, we study the relation between the almost quantum set and the
hierarchy of supersets to the quantum set due to Navascu\'es-Pironio-Ac\'in
(NPA) [Phys. Rev. Lett. 98, 010401 (2007)]. While the former lies between the
first and the third level of the NPA hierarchy, we find that its second level
does not contain and is not contained within the almost quantum set. Finally,
we investigate the hierarchy of refinements to the almost quantum set due to
Moroder et al. [Phys. Rev. Lett. 111, 030501 (2013)], which converges to the
set of quantum correlations producible by quantum states having positive
partial transposition. This allows us to consider (approximations of) the
"biseparable" subsets of the almost-quantum set as well as of the quantum set
of correlations, and thereby gain further insights on the subtle similarities
and differences between the two sets. In addition, they allow us to identify
candidate Bell-like inequalities that can serve as device-independent witnesses
for genuine tripartite entanglement.
