How to deduce the joint distribution of two different elliptically contoured distribution. See details in following.

Question

if $X \sim EC(0,\Sigma_1, g_1)$ and $Y\sim EC(0,\Sigma_2,g_2)$, then what is their joint distribution？That is, let $Z=(X,Y)^T$, what might the distribution of $Z$ be? Clearly, when $X$ and $Y$ are independent, density of $Z$ is $$ p_{\mathbf{z}}(\mathbf{x},\mathbf{y}) = c_{g_{1}}*c_{g_{2}}*|\Sigma_{1}|^{-\frac{1}{2}}*|\Sigma_{2}|^{-\frac{1}{2}}*e^{g_{1}(~\mathbf{x}^{T}\Sigma_{1}^{-1} \mathbf{x}~)+g_{2}(~\mathbf{y}^{T}\Sigma_{2}^{-1} \mathbf{y}~)} $$ But what if $X$ and $Y$ is correlated, assuming that we know the covariance matrix of $X$ and $Y$? Is their any kind of "generalized" elliptically contoured distribution that is used to discribe such random variable $Z$, which consists of different elliptically contoured distributions with different generator function $g(\cdot)$?