This paper presents a purely Lagrangian approach for the 3D simulation of Bingham free-surface fluids and their interaction with deformable solid structures. In the proposed numerical strategy, the fluid is handled using the Particle Finite Element Method (PFEM) to tackle the issues resulting from extreme changes of geometry, such as mesh distortion and free-surface evolution. Additionally, the Papanastasiou model is employed as a regularization technique to overcome the computational difficulties associated with the classical Bingham model. The solid structure, on the other hand, is represented by the hypoelastic constitutive model and simulated using the conventional Finite Element Method (FEM). The coupling between the fluid and the structure is achieved via a monolithic approach, called Unified formulation. Several numerical examples are presented to illustrate the correctness and the robustness of the proposed formulation, in 2D and in 3D. Special attention is devoted to the analysis of the convergence behavior of the proposed computational framework, the effect of the regularization on the numerical results and the 3D effects. Moreover, detailed comparisons between the simulated results and experimental data are performed so that the concerned problems and results can serve as benchmarks.