This paper studies a model of two-phase flow with an immersed material viscous interface and a finite element method for the numerical solution of the resulting system of PDEs. The interaction between the bulk and surface media is characterized by no-penetration and slip with friction interface conditions. The system is shown to be dissipative, and a model stationary problem is proved to be well-posed. The finite element method applied in this paper belongs to a family of unfitted discretizations. The performance of the method when model and discretization parameters vary is assessed. Moreover, an iterative procedure based on the splitting of the system into bulk and surface problems is introduced and studied numerically.