For coupled multiphysics problems, partitioned schemes are often among the few approaches that can accommodate heterogeneous discretization and solver strategies. However, in the presence of strong coupling effects, they struggle to achieve robust coupling between the different physical fields. To address this limitation, we present a framework to construct monolithic multigrid preconditioners for fully coupled systems from pre-existing multigrid-based solvers. Taking a conceptually similar approach as in [1, 2], we combine matrix-free geometric multigrid methods for high-order discretizations in one field with sparse matrix-based algebraic multigrid (AMG) for low-order ones in another. Exploiting the block structure of the system, we construct a monolithic multigrid hierarchy in which the transfer operators and smoothers are composed from the single-field components and the coupling is preserved throughout all multigrid levels. Our implementation is based on two openly available and widely used software packages, the matrix-free infrastructure of the deal.II finite element library [3], and the MueLu AMG [4] framework. We are able to incorporate the performance advantages and parallel scalability of the individual methods while achieving mesh- and polynomial-order-independent iteration counts for the coupled system. In particular, for high-order (p ≥ 3) DG discretizations, we observe a significant reduction in memory usage and solve time compared to available monolithic sparse matrix-based AMG methods and reduced iteration counts when compared to physics-based block preconditioners. We demonstrate the approach for interface problems with different discretization schemes (DG vs. continuous FEM) and polynomial orders for the individual fields across nonmatching meshes.