  The gravitational wave (GW) antenna LISA will detect the signal from
coalescing massive black hole binaries (MBHBs) of $\rm 10^4\,{-}\,10^7\,
M_{\odot}$, providing clues on their formation and growth along cosmic history.
Some of these events will be localized with a precision of several to less than
a deg$^2$, enabling the possible identification of their host galaxy. This work
explores the properties of the host galaxies of LISA MBHBs below
$z\,{\lesssim}\,3$. We generate a simulated lightcone by using the
semi-analytical model $\mathrm{\texttt{L-Galaxies}}$ applied on the merger
trees of the high-resolution N-body cosmological simulation
$\mathrm{\texttt{Millennium-II}}$. The model shows that LISA MBHBs are expected
to be found in optically dim ($r\,{>}\,20$), star-forming ($\rm
sSFR\,{>}\,10^{-10}\, \rm yr^{-1}$), gas-rich ($f_{\rm gas}\,{>}\,0.6$) and
disc-dominated ($\rm B/T\,{<}\,0.7$) \textit{low-mass galaxies} of stellar
masses $10^8\,{-}\,10^9 M_{\odot}$. However, these properties are
indistinguishable from those of galaxies harboring single massive black holes
with comparable mass, making difficult the selection of LISA hosts among the
whole population of low-mass galaxies. Motivated by this, we explore the
possibility of using merger signatures to select LISA hosts. We find that
40-80% of the galaxies housing LISA MBHBs display merger features related to
the interaction which brought the secondary MBH to the galaxy. Despite this,
around 60% of dwarf galaxies placed in the surroundings of the LISA hosts will
show such kind of features as well, challenging the unequivocal detection of
LISA hosts through the search of merger signatures. Consequently, the detection
of an electromagnetic transient associated with the MBHB merger will be vital
to pinpoint the star-forming dwarf galaxy where these binary systems evolve and
coalesce.
