For mildly nonlinear systems involving only diagonal nonlinearities, the global, and essentially monotone, convergence of Newton's method can be established based on concavity/convexity arguments and provided that the Jacobian of the system has a nonnegative inverse. We show that the nonlinear preconditioning resulting from the multi-splitting of the system can preserve those attractive theoretical properties. Our convergence analysis applies in particular to block Jacobi-Newton, RASPEN and the two-step RAS/Newton method. The numerical experiment, based on a discrete porous media equation, shows that the performance of those methods is essentially independent of the mesh size.