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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.