Modelling multiscale-multiphysics geology at field scales is non-trivial due to computational resources and data availability. At such scales it is common to use implicit modelling approaches as they remain a practical method of understanding the first order processes of complex systems. In this work we introduce a numerical framework for the simulation of geomechanical dual-continuum materials. Our framework is written as part of the open source MATLAB Reservoir Simulation Toolbox (MRST). We discretise the flow and mechanics problems using the finite volume method (FVM) and virtual element method (VEM) respectively. The result is a framework that ensures local mass conservation with respect to flow and is robust with respect to gridding. Solution of the coupled linear system can be achieved with either fully coupled or fixed-stress split solution strategies. We demonstrate our framework on an analytical comparison case and on a 3D geological grid case. In the former we observe a good match between analytical and numerical results, for both fully coupled and fixed-stress split strategies. In the latter, the geological model is gridded using a corner point grid that contains degenerate cells as well as hanging nodes. For the geological case, we observe physically plausible and intuitive results given the boundary conditions of the problem. Our initial testing with the framework suggests that the FEM-VEM discretisation has potential for conducting practical geomechanical studies of multiscale systems.