We have performed a layer-resolved, microscopic study of interactions between domain walls in two magnetic layers separated by a non-magnetic one, using high-resolution x-ray photoemission electron microscopy. Domain walls in the hard magnetic Co layer of a Co/Al2O3/FeNi trilayer with in-plane uniaxial anisotropy strongly modify the local magnetization direction in the soft magnetic FeNi layer. The stray fields associated to the domain walls lead to an antiparallel coupling between the local Co and FeNi moments. For domain walls parallel to the easy magnetization axis this interaction is limited to the domain wall region itself. For strongly charged (head-on or tail-to-tail) walls, the antiparallel coupling dominates the interaction over radial distances up to several micrometers from the centre of the domain wall. ; Comment: Published version, J. Phys.: Condens. Matter 19, 476204 (2007)