Multiphase flow in carbonate reservoirs has been a hard problem of scientific research for many years. Accurate flow simulation is essential for the efficient exploitation. In this paper, a coupled triple-continuum and discrete fracture network approach is developed for modeling multiphase flow through fractured vuggy porous media. Multiple levels of fractures can be not only modeled as different superimposed continua but also embodied as discrete fracture network based on their geometrical characteristics. We develop a systematic coupling using Multiscale Finite Element Method (MsFEM) as a framework for coarsening and refinement. MsFEM is used to capture subgrid scale heterogeneities and interactions through multiscale basis functions calculated based on the triple-continuum background. Unstructured mesh is applied to model discrete fractures in arbitrary. This paper presents a significant advancement in terms of elevating the limitations of the triple-continuum models in handling complex fractures and extending the model reduction capability of MsFEM. Several numerical examples are carried out to demonstrate the capability of the proposed coupling method.