We prove a substructuring result for a variational inequality concerning - but not restricted to - the Richards equation in homogeneous soil and including boundary conditions of Signorini's type. This generalizes existing results for the linear case and leads to interface conditions known from linear variational equalities: continuity of Dirichlet and flux values in a weak sense. In case of the Richards equation these are the continuity of the physical pressure and of the water flux, which is hydrologically reasonable. Therefore, we also apply these interface conditions in the heterogeneous case of piecewise constant soil parameters, which we address by the Robin method. We prove that, for a certain time discretization, the homogeneous problems in the subdomains including Robin and Signorini-type boundary conditions can be solved by convex minimization. As a consequence we are able to apply monotone multigrid in the discrete setting as an efficient and robust solver for the local problems. Numerical results demonstrate the applicability of our approach.