Since the birth of computational and data grids, many middlewares have been developed and deployed. Currently, they are used in a multitude of isolated e-Infrastructures and a hot topic in grid technology in the last year or so has been that of middleware interoperability/interoperation. In this paper we present a new approach to grid interoperation based on the so called middleware co-existence. Following this approach, different middlewares are deployed on the same infrastructure and allow the same users to access and/or share the resources, with well defined policy, regardless the middleware they want to use. Although this approach has been used for educational purposes in a training infrastructure, it is so general that it could be used everywhere the interoperation is a concern for the users and/or site managers of an e-Infrastructure.