We experimentally investigate the synchronization dynamics of a self-excited thermoacoustic system forced beyond its phase-locked state. The system consists of a laminar premixed flame in a tube combustor subjected to periodic acoustic forcing. On increasing the forcing amplitude above that required for phase locking, we find that the system can transition out of phase locking and into chaos, which is consistent with the Afraimovich-Shilnikov theorem for the breakdown of a phase-locked torus. However, we also find some unexpected behavior, most notably the emergence of a strange nonchaotic attractor (SNA) before the onset of chaos. We verify the existence of the SNA and chaotic attractor by examining the correlation dimension, the autocorrelation function, the power-law scaling in the Fourier amplitude spectrum, the permutation entropy in a pseudoperiodic surrogate test, and the permutation spectrum. In summary, this study explores the SNA and chaotic dynamics of a thermoacoustic system forced beyond its phase-locked state, opening up new pathways for the development of alternative strategies to control self-excited thermoacoustic oscillations in combustion devices such as gas turbines and rocket engines.