Applying magnetic exchange force microscopy with an Fe-coated tip, we experimentally resolve the atomic-scale antiferromagnetic structure of the Fe monolayer on W(001). On the basis of first-principles calculations, using an Fe nanocluster as a tip, we determine the distance dependence of the magnetic exchange forces. Significant relaxation of tip and sample atoms occurs, which depend sensitively on the local magnetic configuration. This shifts the onset of magnetic interactions toward larger separations and facilitates their observation. Implementing a multiatom tip in the calculations and accounting for relaxation effects are crucial to obtain the correct sign and distance dependence of the magnetic exchange interaction. By comparison with our calculations, we show that the experimentally observed contrast is due to a competition between chemical and magnetic forces.