Once the clusters are generated, there are two types of non-local updates that can be made to the clock states independently in the clock clusters (and thus the spin states in both replicas). First, for every blue cluster, we can flip (or rotate 180 ∘ ∘ ) the clock states with some arbitrary probability. Following this, for every grey cluster (blue clusters connected with red bonds), we can rotate all the clock states simultaneously by a random angle.