Implicit solution methods for multiphysics problems are either monolithic (treating the coupled problem directly) or split (solving reduced systems independently). Multilevel methods must be employed for algorithmic scalability and the block preconditioners developed for split methods can be used as smoothers for coupled multigrid. Monolithic methods require a hierarchical approximation to the coupled system, but the required number of iterations may be smaller due to coupling on all levels. Software is typically structured differently to support ``multigrid inside splitting'' versus ``splitting inside multigrid'', so there are few direct comparisons of the relative effectiveness of these approaches. When splitting is used as a smoother, there is also opportunity to transform the splitting into a multiplicative relaxation scheme similar to ``distributed relaxation''. We describe generic software support for these approaches and compare effectiveness for multiphysics applications in glaciology and geodynamics.