Europe has set an ambitious target to increase the offshore wind power capacity to approximately 30 GW by 2026. With nearshore locations already allocated, future wind farms must be installed in deeper waters, pushing the operational limits of currently used jack-up vessels. Utilizing existing floating heavy-lift vessels presents a viable alternative. This paper disseminates data gathered during the full-scale testing campaign of a floating installation of an offshore wind turbine tower. For this purpose, novel time-synchronized motion-tracking units were developed. Analysis of the obtained data reveals that approximately 96% of the motion response of the tower is due to wave action and 3% to vortex-induced vibrations caused by the presence of a passive tugger line, which shifted one of the system's natural frequencies towards the tower's vortex-shedding frequency. Next to wind and wave-induced motion, the data reveal that the hoisting itself induces tower vibrations, accounting for less than 1% of the tower motion response. The collected data offer a distinctive perspective on this type of installation, which is unlikely to be replicated at model scale due to the scaling limitations associated with the interdependence of waves and wind. The data can be used to validate motion control strategies to enhance the efficiency, safety, and workability of floating offshore wind turbine installations.