  Intermediate-Mass Black Holes (IMBHs) of $10^3-10^6 \, M_\odot$ are commonly
found at the center of dwarf galaxies. Simulations and observations
convincingly show that a sizable population of IMBHs could wander off-center in
galaxies. We use the cosmological simulation ASTRID to study the orbital and
radiative properties of wandering IMBHs in massive galaxies at $z\sim3$. We
find that this population of black holes has large orbital inclinations
($60^\circ\pm22^\circ$) with respect to the principal plane of the host. The
eccentricity of their orbits is also significant ($0.6\pm0.2$) and decreases
with time. Wandering IMBHs undergo spikes of accretion activity around the
pericenter of their orbits, with rates $10^{-3}-10^{-5}$ times the Eddington
rate and a median accretion duty cycle of $\sim 12\%$. Their typical spectral
energy distribution peaks in the infrared at $\sim 11 \, \mu \rm m$ rest-frame.
Assuming a standard value of $10\%$ for the matter-to-energy radiative
efficiency, IMBHs reach $2-10$ keV X-ray luminosities $>10^{37} \,
\mathrm{erg\,s^{-1}}$ for $\sim10\%$ of the time. This luminosity corresponds
to fluxes $>10^{-15} \, \mathrm{erg \, s^{-1} \, cm^{-2}}$ within $10$ Mpc.
They could be challenging to detect because of competing emissions from X-ray
binaries and the interstellar medium. X-ray luminosities $> 10^{41} \,
\mathrm{erg \, s^{-1}}$, in the hyper-luminous X-ray sources (HLXs) regime, are
reached by $\sim 7\%$ of the IMBHs. These findings suggest that HLXs are a
small subset of the wandering IMBH population, which is characterized by
luminosities $10^3-10^4$ times fainter. Dedicated surveys are needed to assess
the demographics of this missing population of black holes.
