  Cluster integrable systems are a broad class of integrable systems modelled
on bipartite dimer models on the torus. Many discrete integrable dynamics arise
by applying sequences of local transformations, which form the cluster modular
group of the cluster integrable system. This cluster modular group was recently
characterized by the first author and Inchiostro. There exist some discrete
integrable dynamics that make use of non-local transformations associated with
geometric $R$-matrices. In this article we characterize the generalized cluster
modular group -- which includes both local and non-local transformations -- in
terms of extended affine symmetric groups. We also describe the action of the
generalized cluster modular group on the spectral data associated with cluster
integrable systems.
