Say $f(X_1)$, $f(X_2)$, $f(X_3)$, $f(X_4)$ are the empirical marginal PDFs of random variables $X_1$, $X_2$ , $X_3$, $X_4$. Also given is correlation between each pair of variables $X_1$, $X_2$ , $X_3$, $X_4$. Is it possible to obtain the empirical joint PDF of $f(X_1, X_2, X_3, X_4)$?
In general, no.
In fact, even if the marginals are normally distributed, the joint distribution could be quite interesting. Here is an one example: Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian?.