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)
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