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This work presents the results of experimental operation of a solar-driven climate system using mixed-integer nonlinear model predictive control (MPC). The system is installed in a university building and consists of two solar thermal collector fields, an adsorption cooling machine with different operation modes, a stratified hot water storage with multiple inlets and outlets as well as a cold water storage. The system and the applied modeling approach is described and a parallelized algorithm for mixed-integer nonlinear MPC and a corresponding implementation for the system are presented. Finally, we show and discuss the results of experimental operation of the system and highlight the advantages of the mixed-integer nonlinear MPC application.