Numerical simulation of fluid–structure interaction (FSI) problems involving multiphase flows remains a challenging task, particularly in the presence of large interface deformations and topological changes such as fluid separation, coalescence, and impact phenomena. Classical moving-mesh approaches, including ALE and space–time formulations, are well established for FSI problems but are inherently limited when fluid–fluid interfaces undergo topological changes. On the other hand, fixed-mesh Eulerian methods overcome this limitation by introducing interface-capturing techniques, typically relying on the advection of indicator or signed-distance functions, whose extension to more than two fluid phases significantly increases algorithmic complexity and computational cost. Within this framework, this work proposes a fully monolithic formulation for the interaction between incompressible multiphase flows and structures, combining a position-based Particle Finite Element Method (PFEM) [1] for the fluid phases with a position-based Finite Element Method (FEM) for large-displacement solid mechanics. Owing to the Lagrangian description inherent to PFEM, fluid particles naturally transport material markers identifying each phase, allowing the direct and cost-free tracking of fluid interfaces, even in the presence of multiple immiscible fluids and topological changes. No additional interface-capturing or advection equations are required. To control the fluid mesh quality, avoiding elements with very small volume, a particle relocation strategy is applied to each fluid phase separately. Both fluid and solid subproblems are formulated using positions as primary nodal variables, which enables a strongly coupled, monolithic FSI formulation within a single nonlinear system of equations. This strategy avoids the convergence difficulties and numerical instabilities commonly observed in staggered or partitioned FSI schemes, particularly in strongly coupled scenarios. The proposed formulation is validated through representative numerical examples involving multiphase flows interacting with flexible structures, demonstrating its robustness for problems characterized by complex interface dynamics and large structural deformations.