This work elaborates on the numerical solution to cope with the enforcement of a general robin-type weak/strong discontinuity interface in acoustic problems, where energy dissipative media with complex-valued properties are involved. We propose a discrete variational formulation based on the idea of Nitsche’s method in the eXtended Finite Element Method (X-FEM) framework. The formulation is stable in the sense that it is able to treat any interface parameters accurately within any mesh configuration and approximation order. We prove the stability of the method, fully considering the effects of energy dissipation, and provide a robust strategy involving basis conversion to obtain the stability parameter. Numerical examples demonstrate that the proposed discrete formulation gives an optimal solution in terms of accuracy and conditioning, compared to a penalty-like formulation. For all interface types, optimal rates of convergence are achieved on both primal and dual variables. Our method exhibits high robustness and stability regardless of discretization, geometries, materials, excitation frequency and interface parameters.