High-redshift QSO spectra show large spatial fluctuations in the Ly-alpha 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-alpha opacity distribution at 5<z<6. The simulation also reproduces the low Thomson scattering optical depth reported by the latest 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-alpha emission by galaxies at z>~6. Reionization is complete at z=5.3 in our model, and 50% of the volume of the Universe is ionized at z=7. Agreement with the Ly-alpha 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 photoionisation 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.