  Galactic outflows are known to consist of several gas phases, however, so far
the connection between these multiple phases has been investigated little and
only in a few objects. In this paper, we analyse MUSE/VLT data of 26 local
(U)LIRGs and study their ionised and neutral atomic phases. We also include
objects from the literature to obtain a total sample of 31 galaxies with
spatially resolved multi-phase outflow information. We find that the ionized
phase of the outflows has on average an electron density three times higher
than the disc ($n_{\rm e, disc}$ $\sim$ 145 cm$^{-3}$ vs $n_{\rm e, outflow}$
$\sim$ 500 cm$^{-3}$), suggesting that cloud compression in the outflow is more
important that cloud dissipation. We find that the difference in extinction
between outflow and disc correlates with the outflow gas mass. Together with
the analysis of the outflow velocities, this suggests that at least some of the
outflows are associated with the ejection of dusty clouds from the disc. This
may support models where radiation pressure on dust contributes to driving
galactic outflows. The presence of dust in outflows is relevant for potential
formation of molecules inside them. We combine our data with millimetre data to
investigate the molecular phase. We find that the molecular phase accounts for
more than 60 $\%$ of the total mass outflow rate in most objects and this
fraction is higher in AGN-dominated systems. The neutral atomic phase
contributes of the order of 10 $\%$, while the ionized phase is negligible. The
ionized-to-molecular mass outflow rate declines slightly with AGN luminosity,
although with a large scatter.
