High-redshift quasi-stellar object (QSO) spectra show large spatial fluctuations in the Ly α opacity of the intergalactic medium on surprisingly large scales at |$z$| ≳ 5.5. We present a radiative transfer simulation of cosmic reionization driven by galaxies that reproduces this large scatter and the rapid evolution of the Ly α opacity distribution at 5 < |$z$| < 6. The simulation also reproduces the low Thomson scattering optical depth reported by the latest cosmic microwave background (CMB) measurement and is consistent with the observed short near-zones and strong red damping wings in the highest redshift QSOs. It also matches the rapid disappearance of observed Ly α emission by galaxies at |$z$| ≳ 6. Reionization is complete at |$z$| = 5.3 in our model, and 50 per cent of the volume of the Universe is ionized at |$z$| = 7. Agreement with the Ly α forest data in such a late reionization model requires a rapid evolution of the ionizing emissivity of galaxies that peaks at |$z$| ∼ 6.8. The late end of reionization results in a large scatter in the photoionization rate and the neutral hydrogen fraction at redshifts as low as |$z$| ≲ 5.5 with large residual neutral ‘islands’ that can produce very long Gunn–Peterson troughs resembling those seen in the data.