  We investigate the impact of quasars on the ionization of the surrounding
intergalactic medium (IGM) with the radiative transfer code \texttt{CRASH4},
now accounting for X-rays and secondary electrons. After comparing with
analytic solutions, we post-process a cosmic volume ($ \approx 1.5\times 10^4$
Mpc$^3 h^{-3}$) containing a ULAS J1120+0641-like quasar (QSO) hosted by a $5
\times 10^{11} {\rm M}_\odot h^{-1}$ dark matter (DM) halo. We find that: (i)
the average HII region ($R\sim3.2$~pMpc in a lifetime $t_f = 10^7$~yrs) is
mainly set by UV flux, in agreement with semi-analytic scaling relations; (ii)
a largely neutral ($x_{\textrm{HII}} < 0.001$), warm ($T\sim 10^3$~K) tail
extends up to few Mpc beyond the ionization front, as a result of the X-ray
flux; (iii) LyC-opaque inhomogeneities induce a line of sight (LOS) scatter in
$R$ as high as few physical Mpc, consistent with the DLA scenario proposed to
explain the anomalous size of the ULAS J1120+0641 ionized region. On the other
hand, with an ionization rate $\dot{N}_{\gamma,0} \sim 10^{57}$~s$^{-1}$, the
assumed DLA clustering and gas opacity, only one LOS shows an HII region
compatible with the observed one. We deduce that either the ionization rate of
the QSO is at least one order of magnitude lower or the ULAS J1120+0641 bright
phase is shorter than $10^7$~yrs.
