Let $\mu,\nu$ be two probability measures on a measurable space $(X,\mathscr A)$. The coupling of $\mu$ and $\nu$ consists of constructing a new probability space $(\Omega,\mathscr F,\mathsf P)$ together with two random variables $$ \begin{align} \xi:(\Omega,\mathscr F)&\to(X,\mathscr A)\quad \\ \eta:(\Omega,\mathscr F)&\to(X,\mathscr A) \end{align} $$ such that $\xi_*(\mathsf P) = \mu_i$ and $\eta_*(\mathsf P) = \nu$. I.e. for example $\mathsf P(\xi^{-1}(A)) = \mu(A)$ for any $A\in \mathscr A$.
I wonder if there are any sufficient/necessary conditions on $\mu,\nu$ which assure that no matter which coupling is chosen, $\xi\perp \eta$ in the sense that $$ \mathsf P(\xi^{-1}(A)\cap \eta^{-1}(B)) = \mu(A)\nu(B). $$