  We present an exact mathematical framework able to describe site-percolation
transitions in real multiplex networks. Specifically, we consider the average
percolation diagram valid over an infinite number of random configurations
where nodes are present in the system with given probability. The approach
relies on the locally treelike ansatz, so that it is expected to accurately
reproduce the true percolation diagram of sparse multiplex networks with
negligible number of short loops. The performance of our theory is tested in
social, biological, and transportation multiplex graphs. When compared against
previously introduced methods, we observe improvements in the prediction of the
percolation diagrams in all networks analyzed. Results from our method confirm
previous claims about the robustness of real multiplex networks, in the sense
that the average connectedness of the system does not exhibit any significant
abrupt change as its individual components are randomly destroyed.
