Mono layers of nanosized carboxylate-functionalized polystyrene particles at the air/water interface assemble into two coexisting liquid phases of different densities. The rheological properties of the individual phases and of the resulting nanoparticle texture is measured using a magnetic needle microrheometer. We show that, despite the liquid behavior of the individual viscous phases, the entire structure exhibits elastic response to local deformations induced by the needle. We explain this response by microscopically nonresolved nanofacets that connect the domains and form a two-dimensional foam.