Integrated energy systems have recently gained primary importance in clean energy transition. The combination of the electricity, heating and gas sectors can improve the overall system efficiency and integration of renewables by exploiting the synergies among the energy vectors. In particular, real-time optimization tools based on Model Predictive Control (MPC) can considerably improve the performance of systems with several conversion units and distribution networks by automatically coordinating all interacting technologies. Despite the relevance of several simulation studies on the topic, however, it is significantly harder to have an experimental demonstration of this improvement. This work presents a methodology for the real-world implementation of a novel smart control strategy for integrated energy systems, based on two coordinated MPC levels, which optimize the operation of all conversion units and all energy vectors in the short- and long-term, respectively, to account also for economic incentives on critical units. The strategy that was previously developed and evaluated in a simulation environment has now been implemented, as a supervisory controller, in the integrated energy system of a hospital in Italy. The optimal control logic is easily actuated by dynamically communicating the optimal set-points to the existing Building Management System, without having to alter the system configuration. Field data collected over a two-year period, firstly when it was business as usual and when the new operation was introduced, show that the MPC increased the economic margin and revenues from yearly incentives and lowered the amount of electricity purchased, reducing dependency on the power grid.