Many-body interactions can couple electronic states in one layer with collective excitations in the adjacent layer, providing a route to tailor properties of heterostructures. However, detecting and quantifying interlayer many-body interactions proved a major challenge. Here, we demonstrate that quasiparticles in monolayer transition metal dichalcogenides (TMDs) are dressed by a remote cloud of phonons in the adjacent hexagonal boron nitride slab. Using angle resolved photoemission, we identify replica bands in the TMDs which are a clear fingerprint of long-range electron-phonon interaction. We develop a modified Fröhlich model that shows semi-quantitative agreement with the experimental spectral functions. Our analysis shows that remote electron-phonon coupling is a generic property of interfaces with hBN. This has implications for electron mobilities in 2D materials, for superconductivity and possibly for moiré correlated phases.