In the context of AdS/CFT, the program of bulk reconstruction aims to reconstruct observables in AdS gravity (or 'bulk' observables) in terms of CFT data. A free scalar field in AdS for example can be reconstructed in terms of smeared primary operators of the CFT in a process called HKLL bulk reconstruction. It is significantly harder to reconstruct interacting fields in AdS, in particular gravitationally interacting fields. In the highly symmetric case of 3-dimensional AdS however, bulk reconstruction of a gravitationally dressed scalar field is possible. In this talk I will discuss how these bulk objects are special from the point of view of CFT entanglement data, in particular what the action of the modular Hamiltonian on the (respectively free or gravitationally interacting) bulk field is. The other way around, the action of the modular Hamiltonian on the CFT representation of the bulk field constrains the field's bulk equation of motion to be consistent with diffeomorphism invariance. The talk will be based on 2205.15244.