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In the protogalactic density field, diffuse gas and collision-less cold dark matter (DM) are often assumed sufficiently mixed that both components experience identical tidal torques. However, haloes in cosmological simulations consistently end up with a higher specific angular momentum (sAM) in gas, even in simulations without radiative cooling and galaxy formation physics. We refine this result by analysing the spin distributions of gas and DM in $\sim$50,000 well-resolved haloes in a non-radiative cosmological simulation from the SURFS suite. The sAM of the halo gas on average ends up $\sim$40\% above that of the DM. This can be pinned down to an excess AM in the inner halo ($<$50\% virial radius), paralleled by a more coherent rotation pattern in the gas. We uncover the leading driver for this AM difference through a series of control simulations of a collapsing ellipsoidal top-hat, where gas and DM are initially well mixed. These runs reveal that the pressurised inner gas shells collapse more slowly, causing the DM ellipsoid to spin ahead of the gas ellipsoid. The arising torque generally transfers AM from the DM to the gas. The amount of AM transferred via this mode depends on the initial spin, the initial axes ratios and the collapse factor. These quantities can be combined in a single dimensionless parameter, which robustly predicts the AM transfer of the ellipsoidal collapse. This simplistic model can quantitatively explain the average AM excess of the gas found in the more complex non-radiative cosmological simulation.