  LISA will extend the search for gravitational waves (GWs) at $0.1\,{-}\,100$
mHz where loud signals from coalescing binary black holes of $ 10^4
\,{-}\,10^7\,\rm M_{\odot}$ are expected. Depending on their mass and
luminosity distance, the uncertainty in the LISA sky-localization decreases
from hundreds of deg$^2$ during the inspiral phase to fractions of a deg$^2$
after the merger. By using the semi-analytical model L-Galaxies applied to the
Millennium-I merger trees, we generate a simulated Universe to identify the
hosts of $z\,{\leq}\,3$ coalescing binaries with total mass of
$3\,{\times}\,10^{5}$, $3\,{\times}\,10^6$ and $3\,{\times}\,10^7\rm
M_{\odot}$, and varying mass ratio. We find that, even at the time of merger,
the number of galaxies around the LISA sources is too large (${\gtrsim}\,10^2$)
to allow direct host identification. However, if an X-ray counterpart is
associated to the GW sources at $z\,{<}\,1$, all LISA fields at merger are
populated by ${\lesssim}\,10$ AGNs emitting above ${\sim}\, 10^{-17} \, \rm
erg\,cm^{-2}\,s^{-1}$. For sources at higher redshifts, the poorer
sky-localization causes this number to increase up to ${\sim}\, 10^3$. Archival
data from eRosita will allow discarding ${\sim}\, 10\%$ of these AGNs, being
too shallow to detect the dim X-ray luminosity of the GW sources. Inspiralling
binaries in an active phase with masses ${\lesssim}\,10^6\rm M_{\odot}$ at
$z\,{\leq}\,0.3$ can be detected, as early as $10$ hours before the merger, by
future X-ray observatories in less than a few minutes. For these systems,
${\lesssim}\,10$ AGNs are within the LISA sky-localization area. Finally, the
LISA-Taiji network would guarantee the identification of an X-ray counterpart
$10$ hours before merger for all binaries at $z\,{\lesssim}\,1$.
