This paper presents the investigation of high-frequency thermoacoustic limit-cycle oscillations in a novel experimental gas turbine reheat combustor featuring both auto-ignition and propagation stabilised flame zones at atmospheric pressure. Dynamic pressure measurements at the faceplate of the reheat combustion chamber reveal high-amplitude periodic pressure pulsations at 3 kHz in the transverse direction of the rectangular cross-section combustion chamber. Further analysis of the acoustic signal shows that this is a thermoacoustically unstable condition undergoing limit-cycle oscillations. A sensitivity study is presented which indicates that these high-amplitude limit-cycle oscillations only occur under certain conditions: namely high power settings with propane addition to increase auto-ignition propensity. The spatially-resolved flame dynamics are then investigated using CH* chemiluminescence, phase-locked to the dynamic pressure, captured from all lateral sides of the reheat combustion chamber. This reveals strong heat release oscillations close to the chamber walls at the instability frequency, as well as axial movement of the flame tips in these regions and an overall transverse displacement of the flame. Both the heat release oscillations and the flame motion occur in phase with the acoustic mode. From these observations, likely thermoacoustic driving mechanisms which lead to the limit-cycle oscillations are inferred. In this case, the overall flame-acoustics interaction is assumed to be a superposition of several effects, with the observations suggesting strong influences from autoignition-pressure coupling as well as flame displacement and deformation due to the acoustic velocity field. These findings provide a foundation for the overall objective of developing predictive approaches to mitigate the impact of high-frequency thermoacoustic instabilities in future generations of gas turbines with sequential combustion systems.