The problem of computing in an efficient and accurate way the armor losses in three-core AC submarine cables has been of great interest to the offshore cable community for some years. Indeed, efficiency of the computation method is crucial to allow comprehensive parametric analysis as might be needed in the design study. In (Giussani <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> 2021), an algorithm based on integral equations for performing such calculations with an accuracy comparable (within acceptable tolerances) to state-of-the-art Finite Element Method (FEM) approaches was presented. In the present study, we extend the above mentioned algorithm to account for the mutual inductive coupling between armor wires and metallic screens, which are often present in actual cables structure. We show that the most notable properties of the original algorithm are preserved in this coupled formulation. In particular we discuss how exploiting the cable structure periodicity and symmetry leads to huge computational cost reductions. The new extended simulation tool is applied to a parametric study to assess how geometry and material properties influence losses and to assess in particular the impact of the metallic sheaths.