In this paper a coupled Eulerian-Lagrangian method based on level set formulti-material compressible fluid flow involving large deformation ispresented. In the scheme, a Lagrangian method is used in the domain withsmall deformation while an Euler method is used in the domain with largedeformation. The interface is treated by a new level set method and theGhost Fluid method. Furthermore, an explicit compatible finite elementmethod is employed as the Lagrangian method. In this Lagrangian method, thefinite element method is used to discretize the fluid equations in terms ofthe framework of compatibility, an edge-centered artificial viscosity isused to capture shocks, and the subzonal perturbed pressure is used toresist spurious and unresolved grid motions. A vertex-centered finite volumemethod based on approximate Riemann solver is used as the Eulerian method.In this Eulerian method, an HLLC numerical flux adapted to various equationsof state are employed, a second order accuracy in space is achieved by usinga reconstruction of primitive variables based on WENO approach. An explicittwo-stage Runge-Kutta time-stepping scheme is used in discretization oftime. A new method to determine the signs of the level set function and anew technique to compute the velocities of the projection points of theGhost points on the interface are proposed. Numerical examples show theaccuracy as well as the robustness of the method.