The magnetization reversal processes in an exchange-coupled nanocomposite system, consisting of a continuous film of the ferromagnet Ni 77Fe14Mo15Cu4with a square mesh of the antiferromagnet FeMn on top, as a function of an external magnetic field were investigated by the magneto-optical indicator film technique. It was found that the boundary conditions near the antiferromagnetic (AFM) stripes crucially influence the remagnetization kinetics in the ferromagnetic film. Unusual behavior was revealed for different topologically stable structures at 180° domain walls in the ferromagnetic film. Also, the magnetization reversal of the ferromagnetic regions underneath the antiferromagnetic stripes is different for applied fields parallel and perpendicular to the sample's unidirectional anisotropy. In both cases, an asymmetry in the magnetic structure transformation during remagnetization of the AFM-covered parts of the ferromagnetic film was observed for the forward and backward branches of the hysteresis loop.