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  1. ABSTRACT We present results of MUSE-ALMA haloes, an ongoing study of the circumgalactic medium (CGM) of galaxies (z ≤ 1.4). Using multiphase observations we probe the neutral, ionized, and molecular gas in a subsample containing six absorbers and nine associated galaxies in the redshift range z ∼ 0.3–0.75. Here, we give an in-depth analysis of the newly CO-detected galaxy Q2131−G1 (z = 0.42974), while providing stringent mass and depletion time limits for the non-detected galaxies. Q2131−G1 is associated with an absorber with column densities of log(NH i/cm−2) ∼ 19.5 and $\textrm {log}(N_{\textrm {H}_2}/\textrm {cm}^{-2}) \sim 16.5$, and has a star formationmore »rate of SFR = 2.00 ± 0.20 M⊙yr−1, a dark matter fraction of fDM(r1/2) = 0.24–0.54, and a molecular gas mass of $M_\textrm {mol} = 3.52 ^{+3.95}_{-0.31} \times 10^9 \,\, \textrm {M}_{\odot }$ resulting in a depletion time of τdep < 4.15 Gyr. Kinematic modelling of both the CO (3–2) and [O iii] λ5008 emission lines of Q2131−G1 shows that the molecular and ionized gas phases are well aligned directionally and that the maximum rotation velocities closely match. These two gas phases within the disc are strongly coupled. The metallicity, kinematics, and orientation of the atomic and molecular gas traced by a two-component absorption feature are consistent with being part of the extended rotating disc with a well-separated additional component associated with infalling gas. Compared to emission-selected samples, we find that H i-selected galaxies have high molecular gas masses given their low star formation rate. We consequently derive high depletion times for these objects.« less