  We derive ``Bell inequalities'' in four dimensional phase space and prove the
following ``three marginal theorem'' for phase space densities
$\rho(\overrightarrow{q},\overrightarrow{p})$, thus settling a long standing
conjecture : ``there exist quantum states for which more than three of the
quantum probability distributions for $(q_1,q_2)$, $(p_1,p_2)$, $(q_1,p_2)$ and
$(p_1,q_2)$ cannot be reproduced as marginals of a positive
$\rho(\overrightarrow{q},\overrightarrow{p})$''. We also construct the most
general positive $\rho(\overrightarrow{q},\overrightarrow{p})$ which reproduces
any three of the above quantum probability densities for arbitrary quantum
states. This is crucial for the construction of a maximally realistic quantum
theory.
