The definition of coupling is written below. In some articles, I found the term "natural coupling". When is a coupling said to be ''natural''?
Definition of coupling between two random variables: Let $X_1$ and $X_2$ be two random variables defined on probability spaces $(\Omega_1,F_1,P_1)$ and $(\Omega_2,F_2,P_2)$. Then a coupling of $X_1$ and $X_2$ is a new probability space $(\Omega,F,P)$ over which there are two random variables $Y_1$ and $Y_2$ such that $Y_1$ has the same distribution as $X_1$ while $Y_2$ has the same distribution as $X_2$.