Invariant monotone coupling need not exist

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.
Comment: Published in at http://dx.doi.org/10.1214/12-AOP767 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)