High signal-to-noise observations of the Ly α forest transmissivity in the z = 7.085 quasi-stellar object (QSO) ULAS J1120+0641 show seven narrow transmission spikes followed by a long 240 cMpc h^−1 trough. Here, we use radiative transfer simulations of cosmic reionization previously calibrated to match a wider range of Ly α forest data to show that the occurrence of seven transmission spikes in the narrow redshift range z = 5.85–6.1 is very sensitive to the exact timing of reionization. Occurrence of the spikes requires the most underdense regions of the intergalactic medium to be already fully ionized. The rapid onset of a long trough at z = 6.12 requires a strong decrease of the photoionization rate Γ at z ≳ 6.1 in this line of sight, consistent with the end of percolation at this redshift. The narrow range of reionization histories that we previously found to be consistent with a wider range of Ly α forest data have a reasonable probability of showing seven spikes and the mock absorption spectra provide an excellent match to the spikes and the trough in the observed spectrum of ULAS J1120+0641. Larger samples of high signal-to-noise searches for rare Ly α transmission spikes at z > 5.8 should therefore provide important further insights into the exact timing of the percolation of H ii bubbles at the tail end of reionization.