Many real-world network are multilayer, with nontrivial correlations across layers. Here we show
that these correlations amplify geometry in networks. We focus on mutual clustering—a measure
of the amount of triangles that are present in all layers among the same triplets of nodes—and
find that this clustering is abnormally high in many real-world networks, even when clustering in
each individual layer is weak. We explain this unexpected phenomenon using a simple multiplex
network model with latent geometry: links that are most congruent with this geometry are the ones
that persist across layers, amplifying the cross-layer triangle overlap. This result reveals a different
dimension in which multilayer networks are radically distinct from their constituent layers.