In this work, we compare these two approaches for preconditioning the iterative solution of a coupled multi-field problem derived from a generalized continuum model. The model is aimed at simulating failure in quasi-brittle materials such as concrete and rocks and it couples microrotation and nonlocal damage fields with the displacement field. Solving large and sparse linear systems can be a challenge, especially when dealing with complex systems like multi-field problems. The success of iterative methods depends on the spectral properties of the system matrix, so careful construction of the preconditioner is crucial. A usual approach for the multi-field case involves using a block preconditioner based on factorization, which requires problem-specific approximations of the Schur complement and sub-block inverses. A popular choice for these inverse approximations is the Algebraic Multigrid Method (AMG), which has proven effective in many cases. Another strategy is to use AMG on the entire system as a monolithic preconditioner, which treats the block structure inside the preconditioner during hierarchy construction. Our investigations intend to demonstrate, for the proposed problem, the usefulness of treating the whole block system inside the AMG hierarchy and hence preserve the coupling aspects of the problem.