Composites of magnetically hard and soft phases are present in multiple and diverse applications, ranging from bulk permanent magnets in motors and generators to state-of-the-art recording media devices. The nature of the magnetic coupling between the hard and soft phases is of great technological relevance, as the macroscopic properties of the functional composite material ultimately depend on the atomic-scale interactions between phases. In this work, the hard/soft bilayer system SrFe₁₂O₁₉/Co has been studied based on photoemission electron microscopy combined with x-ray absorption and magnetic circular dichroism. Our experiments show that the magnetization of the hard magnetic oxide has a direction perpendicular to the layer plane, whereas the magnetization of the soft metallic overlayer remains in-plane. As a consequence, the magnetic domain patterns observed for the hard and soft phases are very different and completely uncorrelated to one another, indicating that no soft spins align with the hard phase by pure magnetodipolar arguments. The results are understood as the consequence of an absence of exchange-coupling between phases, in a scenario in which the shape anisotropy of the soft layer overcomes the Zeeman energy of the perpendicular magnetic field generated by the hard ferrite. Micromagnetic simulations of our system predict that low degrees of exchange-coupling effectively prevent substantial softening of the composite and lead to the alignment of soft and hard magnetic moments. A strategy thus emerges for the development of future hard-soft magnets, based on minimizing the degree of exchange-coupling while avoiding complete uncoupling.