  We use a set of AMR hydrodynamic simulations post-processed with the
radiative-transfer code RADAMESH to study how inhomogeneous HeII reionization
affects the intergalactic medium (IGM). We propagate radiation from active
galactic nuclei (AGNs) considering two scenarios for the time evolution of the
ionizing sources. We find that HeII reionization takes place in a very
inhomogeneous fashion, through the production of well separated bubbles of the
ionized phase that subsequently percolate. Overall, the reionization process is
extended in time and lasts for a redshift interval Delta z>1. At fixed gas
density, the temperature distribution is bimodal during the early phases of
HeII reionization and cannot be described by a simple effective equation of
state. When HeII reionization is complete, the IGM is characterized by a
polytropic equation of state with index gamma~1.20. This relation is
appreciably flatter than at the onset of the reionization process (gamma=1.56)
and also presents a much wider dispersion around the mean. We extract HI and
HeII Ly-alpha absorption spectra from the simulations and fit Voigt profiles to
them. We find that the regions where helium is doubly ionized are characterized
by different probability density functions of the curvature and of the Doppler
b parameters of the HI Ly-alpha forest as a consequence of the bimodal
temperature distribution during the early phases of HeII reionization. The
column-density ratio in HeII and HI shows a strong spatial variability. Its
probability density function rapidly evolves with time reflecting the
increasing volume fraction in which ionizing radiation is harder due to the AGN
contribution. Finally we show that the number density of the flux-transmission
windows per unit redshift and the mean size of the dark gaps in the HeII
spectra have the potential to distinguish between different reionization
scenarios. (abridged)
