We show by example that there is a Cayley graph, having two invariant random subgraphs $X$ and $Y$, such that there exists a monotone coupling between them in the sense that $X\subset Y$, although no such coupling can be invariant. Here, “invariant” means that the distribution is invariant under group multiplications.
"Invariant monotone coupling need not exist." Ann. Probab. 41 (3A) 1180 - 1190, May 2013. https://doi.org/10.1214/12-AOP767