  We develop the theory of sparse multiplex networks with partially overlapping
links based on their local tree-likeness. This theory enables us to find the
giant mutually connected component in a two-layer multiplex network with
arbitrary correlations between connections of different types. We find that
correlations between the overlapping and non-overlapping links markedly change
the phase diagram of the system, leading to multiple hybrid phase transitions.
For assortative correlations we observe recurrent hybrid phase transitions.
