  Massive black-hole (BH) mergers are predicted to be powerful sources of
low-frequency gravitational waves (GWs). Coupling the detection of GWs with an
electromagnetic (EM) detection can provide key information about merging BHs
and their environments. We study the high-resolution cosmological
radiation-hydrodynamics simulation OBELISK, run to redshift $z=3.5$, to assess
the GW and EM detectability of high-$z$ BH mergers, modelling spectral energy
distribution and obscuration. For EM detectability, we further consider
sub-grid dynamical delays in postprocessing. We find that most of the merger
events can be detected by LISA, except for high-mass mergers with very unequal
mass ratios. Intrinsic binary parameters are accurately measured, but the sky
localisation is poor generally. Only $\sim 40\%$ of these high-$z$ sources have
a sky localisation better than $10\,\mathrm{deg}^2$. Merging BHs are hard to
detect in the restframe UV since they are fainter than the host galaxies, which
at high $z$ are star-forming. A significant fraction, $15-35\%$, of BH mergers
instead outshine the galaxy in X-rays, and about $5-15\%$ are sufficiently
bright to be detected with sensitive X-ray instruments. If mergers induce an
Eddington-limited brightening, up to $30\%$ of sources can become observable.
The transient flux change originating from such a brightening is often large,
allowing $4-20\%$ of mergers to be detected as EM counterparts. A fraction,
$1-30\%$, of mergers are also detectable at radio frequencies. Observable
merging BHs tend to have higher accretion rates and masses and are overmassive
at a fixed galaxy mass with respect to the full population. Most EM-observable
mergers can also be GW-detected with LISA, but their sky localisation is
generally poorer. This has to be considered when using EM counterparts to
obtain information about the properties of merging BHs and their environment.
