In modern civil engineering, advanced vibration control devices are used to reduce the dynamic response of structures under various loadings such as wind or earthquakes. To eliminate uncertainties due to scaling effects and damper-structure interactions, large-scale experimental dynamic testing of prototypes in realistic structures is necessary. To address these challenges and overcome the limitations of conventional experimental methods, real-time hybrid simulation (RTHS) has emerged as a powerful computation method which combines finite element simulations with experimental testing in the laboratory. The structure is divided into a numerical and an experimental substructure while the displacements and forces at their interface are calculated, measured and synchronized. However, the numerical model has to be solved in real-time in order to capture velocity dependent damping effects, thus limiting the size and complexity of the finite element model. Especially nonlinear calculations, which require repeated calculations in local Newton-Raphson iterations, are a numerical challenge. To overcome this, we propose to implement model order reduction techniques into the RTHS workflow. To cover nonlinear problems, a proper orthogonal decomposition (POD) based linear reduction is combined with the hyper reduction method energy conserving mesh sampling and weighting (ECSW). After a computational expensive offline-phase, this allows for a highly reduced number of degrees of freedom in the online-phase during the RTHS. The system is solved in an optimized, lower-dimensional subspace whereas the nonlinear internal load vector is constructed using a reduced number of elements. In this study, we test two large-scale shape memory alloy based damper prototypes in a realistic, industrial-like steel frame structure under various earthquake excitations using RTHS. The dampers lead to a great reduction of the overall dynamic response of the structure while the combination of POD and ECSW shows great potential for larger and more complex numerical substructures.