Many multi-phase materials contain interfaces exhibiting a peculiar deformation mode, where one of the phases deforms anisotropically by forming serrated, jagged, interface impinging into another (approximately isotropic) phase. Examples include twins impinging on a grain boundary, crystalline-amorphous interface or martensite-ferrite interface in advanced multi-phase steels, where martensite islands typically deform by sliding on the retained austenite films. Such serrated deformation mode induces locally large strain concentrations at the fine scale in the near-interface second phase, where nano-voids can form and grow, leading to damage and crack formation. Classical cohesive zone models are, in general, not able to incorporate such effects. In this contribution, we propose a novel, enhanced multi-phase interface modelling approach that describes interface damage resulting from the jagged sliding of one of the phases. At the mesoscale, a two-phase mesostructure is considered. Microscale model represents an interfacial zone unit cell resolving the substructure and anisotropic, discrete deformation, leading to an enhanced cohesive interface description at the mesoscale. Applying the extended Hill-Mandel condition yields the generalized tractions conjugated to the interface kinematic quantities. Relating the effective quantities, leads to an enhanced cohesive law. This microphysics-based effective interface model is fully identified using a set of ``off-line'' representative unit cell simulations. The developed enhanced interface model is numerically validated against fully resolved modelling of infinite mesoscopic interfacial zones. As an example, the model is applied to the analysis of interface damage in a dual-phase (DP) steel microstructure, where the model performance is validated against microscale experimental results.