Stratified fluids composed of a sequence of alternate layers show interesting macroscopic properties, which may be quite different from those of the individual constituent fluids. On a macroscopic scale, such systems can be considered a sort of fluid metamaterial. In many cases each fluid layer can be described by Euler equations following the stiffened gas equation of state. The computation of detailed numerical solutions of such stratified material poses several challenges, first and foremost the issue of artificial smearing of material parameters across interface boundaries. Lagrangian schemes completely eliminate this issue, but at the cost of rather stringent time step restrictions. In this work we introduce an implicit numerical method for the multimaterial Euler equations in Lagrangian coordinates. The implicit discretization is aimed at bypassing the prohibitive time step restrictions present in flows with stratified media, where one of the materials is particularly dense, or rigid (or both). This is the case for flows of water-air mixtures, air-granular media, or similar high density ratio systems. We will present the novel discretisation approach, which makes extensive use of the remarkable structure of the governing equations in Lagrangian coordinates to find the solution by means of a single implicit discrete wave equation for the pressure field, yielding a symmetric positive definite structure and thus a particularly efficient algorithm. Additionally, we will introduce simple filtering strategies for counteracting the emergence of pressure or density oscillations typically encountered in multimaterial flows, and will present results concerning the robustness, accuracy, and performance of the proposed method, including applications to stratified media with high density and stiffness ratios.