  We present a multi-wavelength study of seven AGN at spectroscopic redshift
>2.5 in the 7 Ms Chandra Deep Field South, selected to have good FIR/sub-mm
detections. Our aim is to investigate the possibility that the obscuration
observed in the X-rays can be produced by the interstellar medium (ISM) of the
host galaxy. Based on the 7 Ms Chandra spectra, we measured obscuring column
densities N$_{H, X}$ in excess of 7x10$^{22}$ cm$^{-2}$ and intrinsic X-ray
luminosities L$_{X}$>10$^{44}$ erg s$^{-1}$ for our targets, as well as
equivalent widths for the Fe K$\alpha$ emission line EW>0.5-1 keV. We built the
UV-to-FIR spectral energy distributions by using broad-band photometry from
CANDELS and Herschel catalogs. By means of an SED decomposition technique, we
derived stellar masses (M$_{*}$~10$^{11}$ Msun), IR luminosities
(L$_{IR}$>10$^{12}$ Lsun), star formation rates (SFR~190-1680 Msun yr$^{-1}$)
and AGN bolometric luminosities (L$_{bol}$~10$^{46}$ erg s$^{-1}$) for our
sample. We used an empirically-calibrated relation between gas masses and
FIR/sub-mm luminosities and derived M$_{gas}$~0.8-5.4x10$^{10}$ Msun.
High-resolution (0.3-0.7'') ALMA data (when available, CANDELS data otherwise)
were used to estimate the galaxy size and hence the volume enclosing most of
the ISM under simple geometrical assumptions. These measurements were then
combined to derive the column density associated with the ISM of the host, on
the order of N$_{H, ISM}$~10$^{23-24}$ cm$^{-2}$. The comparison between the
ISM column densities and those measured from the X-ray spectral analysis shows
that they are similar. This suggests that, at least at high redshift,
significant absorption on kpc scales by the dense ISM in the host likely adds
to or substitutes that produced by circumnuclear gas on pc scales (i.e., the
torus of unified models). The lack of unobscured AGN among our ISM-rich targets
supports this scenario.
