It is shown that large amplitude periodic oscillations of the magnetic field measured frequently by spacecraft in the magnetosheath, and interpreted as mirror-mode waves/structures, represent trains of slow-mode magnetosonic solitons. Nonlinear magnetosonic wave-trains with properties attributed previously to mirror-mode structures are obtained as exact solutions of Hall-MHD equations with anisotropic ion pressure. The theoretically derived properties of magnetosonic solitons are compared with multipoint measurements on Cluster satellites in the magnetosheath with plasma beta (plasma/magnetic pressure) ~10. The computed properties of nonlinear waves: amplitudes, spatial scales, periodicity, and propagation velocities are consistent with in situ measurements in space.