  We demonstrate genuine three-mode nonlocality based on phase space formalism.
A Svetlichny-type Bell inequality is formulated in terms of the
$s$-parameterized quasiprobability function. We test such tool using exemplary
forms of three-mode entangled states, identifying the ideal measurement
settings required for each state. We thus verify the presence of genuine
three-mode nonlocality that cannot be reproduced by local or nonlocal hidden
variable models between any two out of three modes. In our results, GHZ- and
W-type nonlocality can be fully discriminated. We also study the behavior of
genuine tripartite nonlocality under the effects of detection inefficiency and
dissipation induced by local thermal environments. Our formalism can be useful
to test the sharing of genuine multipartite quantum correlations among the
elements of some interesting physical settings, including arrays of trapped
ions and intracavity ultracold atoms.
