In this talk, we are concerned with multifield coupled problems with different, often conflictual types of non-linearities. We bring into focus the challenges of getting numerical solutions. As for instance, we share our investigations of monolithic Newton-multigrid FEM solver for thixotropic flow problems. On one hand, nonlinear multifield coupled problems are often lacking unified FEM analysis due to the presence of different non-linearities. Thus, the importance of treating auxiliary subprob- lems with different analysis tools to guarantee existence of solutions. Moreover, the nonlinear multifield problems are extremely sensitive to the coupling. On other hand, monolithic Newton-multigrid FEM solver show a great success in getting nu- merical solutions for multifield coupled problems. Thixo-viscoplastic flow problem is a perfect example in this regard. It is a two field coupled problem, by means of microstructure dependent plastic-visosity as well as microstructure de- pendent yield stress. Furthermore, the FEM analysis of nonlinear viscoplastic subproblem can only be done using monotone property of the corresponding operators, while the coupled field problems lack such property. In this talk, we show how we adapt the FEM analysis for existence of solutions, as well as for the error analysis for thixo-viscoplastic flow problems. With respect to solver, we show how to develop an efficient monolithic Newton-multigrid solver based on incompressibility constraint in absence of microstructure and pressure coupling. The talk is based on our new results for thixo-viscoplastic flow problems.