  It is commonly assumed in percolation theories that at most one percolating
cluster can exist in a network. We introduce sausage-like networks (SLNs), an
ensemble of synthetic modular networks in which more than one percolating
cluster can appear. We show that coexisting percolating clusters (CPCs) emerge
in such networks due to limited mixing, i.e., a small number of interlinks
between pairs of modules. We develop an approach called modular message passing
(MMP) to describe and verify these observations. We demonstrate that the
appearance of CPCs is an important source of inaccuracy in the previously
introduced percolation theories, such as the message passing (MP) approach.
Moreover, we show that the MMP theory improves significantly over the
predictions of MP for percolation on synthetic networks with limited mixing and
also on several real-world networks. These findings have important implications
for understanding the robustness of networks and in quantifying epidemic
outbreaks in the susceptible-infected-recovered (SIR) model of disease spread.
