  We investigate the genuine multipartite nonlocality of three-mode Gaussian
states of continuous variable systems. For pure states, we present a simplified
procedure to obtain the maximum violation of the Svetlichny inequality based on
displaced parity measurements, and we analyze its interplay with genuine
tripartite entanglement measured via Renyi-2 entropy. The maximum Svetlichny
violation admits tight upper and lower bounds at fixed tripartite entanglement.
For mixed states, no violation is possible when the purity falls below 0.86. We
also explore a set of recently derived weaker inequalities for three-way
nonlocality, finding violations for all tested pure states. Our results provide
a strong signature for the nonclassical and nonlocal nature of Gaussian states
despite their positive Wigner function, and lead to precise recipes for its
experimental verification.
