an incoherent interface

At the opposite extreme from a coherent interface is one where the two crystals simply give up trying to line up. Their structures, or their spacings, or their orientations are so different that there is no row-for-row correspondence across the seam at all; the atoms just meet in a jumbled, poorly-fitted layer, much like a random high-angle grain boundary. This no-matching interface is an incoherent interface.

An incoherent interface has essentially NO continuity of lattice planes across it; the misfit is too large (or the two structures too unrelated) for either elastic strain or a tidy array of misfit dislocations to accommodate. The atoms in the boundary sit in a disordered compromise, so the interface energy is high (comparable to or above a general grain boundary) and does not depend sensitively on orientation. Think of it as the interphase analogue of a random high-angle grain boundary. Because it is disordered, it is also mobile and a fast diffusion path.

Incoherent interfaces are what you get around large, coarsened precipitates, or between phases with completely unrelated crystal structures, or at the boundary of a phase formed with no orientation relationship to its host. Their high energy makes them strong drivers of coarsening (the system wants to reduce total interface area), but per unit area they are less effective at some strengthening tasks than coherent interfaces because they cannot transmit coherency strain. Honest note: incoherent does not mean there is a gap or that the phases are not bonded. The atoms are still bonded across it; they are simply not registered plane-to-plane.