Dedicated tip-tilt loops are commonly implemented on adaptive optics (AO) systems. In addition, a number of recent high-performance systems feature tip-tilt controllers which are more efficient than the integral action controller. In this context, Linear Quadratic Gaussian (LQG) tip-tilt regulators based on stochastic models identified from AO telemetry have demonstrated their capacity to effectively compensate for the cumulated effects of atmospheric disturbance, windshake and vibrations. These tip-tilt LQG regulators can also be periodically re-tuned during AO operations, thus allowing to track changes in the disturbances’ temporal dynamics. This paper investigates the potential benefit of extending the number of low-order modes to be controlled using models identified from AO telemetry. The global stochastic dynamical model of a chosen number of turbulent low-order modes is identified through data-driven modelling from wavefront sensor measurements. The remaining higher modes are modelled using priors with autoregressive models of order 2. The loop is then globally controlled using the optimal LQG regulator build from all these models. Our control strategy allows for combining a dedicated tip-tilt loop with a deformable mirror that corrects for the remaining low-order modes and for the higher orders altogether, without resorting to mode decoupling. Performance results are obtained through evaluation of the Strehl ratio computed on H-band images from the scientific camera, or in replay mode using on-sky AO telemetry recorded in July 2019 on the CANARY instrument.