Diffractions encode subwavelength information and superior illumination but are weak in amplitude and strongly interfere with the more dominant reflected wavefield. Accordingly, the successful separation of diffractions in geophysical applications still forms a major direction of research. Confronting this challenge, an automatable and versatile framework for the separation of interfering wavefields through a sequence of coherent data summation and subtraction is proposed. In contrast to other studies using coherence measurements, the method specifically targets reflected contributions, which are normally favored in automated summation schemes, and it adaptively subtracts the resulting reflection stack from the input data. Complementing this natural selectivity of stacking, a variety of wavefront filters is presented, which can be derived from the local maximization of the semblance norm and additionally inform the subsequent separation step. Complex 2D and 3D synthetic pre- and poststack seismic examples together with a ground-penetrating radar field-data example suggest that the presented scheme can be applied to a variety of data configurations and bears the potential to uncover an extremely faint but surprisingly rich diffraction background that was previously not accessible for dedicated processing.