  The thermal state of the post-reionization IGM is sensitive to the timing of
reionization and the nature of the ionizing sources. We have modelled here the
thermal state of the IGM in cosmological radiative transfer simulations of a
realistic, extended, spatially inhomogeneous hydrogen reionization process,
carefully calibrated with Ly-alpha forest data. We compare these with
cosmological simulations run using a spatially homogeneous ionizing background.
The simulations with a realistic growth of ionized regions and a realistic
spread in reionization redshifts show, as expected, significant spatial
fluctuations in the temperature-density relation (TDR) of the post-reionization
IGM. The most recently ionized regions are hottest and exhibit a flatter TDR.
In simulations consistent with the average TDR inferred from Ly-alpha forest
data, these spatial fluctuations have a moderate but noticeable effect on the
statistical properties of the Ly-alpha opacity of the IGM at z ~ 4-6. This
should be taken into account in accurate measurements of the thermal properties
of the IGM and the free-streaming of dark matter from Ly-alpha forest data in
this redshift range. The spatial variations of the TDR predicted by our
simulations are, however, smaller by about a factor two than would be necessary
to explain the observed large spatial opacity fluctuations on large (> 50
comoving Mpc/h) scales at z > 5.5.
