We show that all nonlocal Hamiltonian structures of hydrodynamic type compatible with a given nondegenerate {\it local} Hamiltonian structure of Dubrovin—Novikov type can be written as the Lie derivatives of the latter along suitably chosen nonlocal vector fields. We also discuss applications of this result to the classification of compatible nonlocal Hamiltonian structures of hydrodynamic type and of the corresponding metrics.