The field of miniature mechanical oscillators is rapidly evolving, with emerging applications including signal processing, biological detection(1) and fundamental tests of quantum mechanics(2). As the dimensions of a mechanical oscillator shrink to the molecular scale, such as in a carbon nanotube resonator(3-7), their vibrations become increasingly coupled and strongly interacting(8,9) until even weak thermal fluctuations could make the oscillator nonlinear(10-13). The mechanics at this scale possesses rich dynamics, unexplored because an efficient way of detecting the motion in real time is lacking. Here we directly measure the thermal vibrations of a carbon nanotube in real time using a high-finesse micrometre-scale silicon nitride optical cavity as a sensitive photonic microscope. With the high displacement sensitivity of 700 fm Hz(-1/2) and the fine time resolution of this technique, we were able to discover a realm of dynamics undetected by previous time-averaged measurements and a room-temperature coherence that is nearly three orders of magnitude longer than previously reported. We find that the discrepancy in the coherence stems from long-time non-equilibrium dynamics, analogous to the Fermi-Pasta-Ulam-Tsingou recurrence seen in nonlinear systems(14). Our data unveil the emergence of a weakly chaotic mechanical breather(15), in which vibrational energy is recurrently shared among several resonance modes-dynamics that we are able to reproduce using a simple numerical model. These experiments open up the study of nonlinear mechanical systems in the Brownian limit (that is, when a system is driven solely by thermal fluctuations) and present an integrated, sensitive, high-bandwidth nanophotonic interface for carbon nanotube resonators.