ABSTRACT Understanding what shapes the cold gas component of galaxies, which both provides the fuel for star formation and is strongly affected by the subsequent stellar feedback, is a crucial step towards a better understanding of galaxy evolution. Here, we analyse the H i properties of a sample of 46 Milky Way halo-mass galaxies, drawn from cosmological simulations (EMP-Pathfinder and Firebox). This set of simulations comprises galaxies evolved self-consistently across cosmic time with different baryonic sub-grid physics: three different star formation models [constant star formation efficiency (SFE) with different star formation eligibility criteria, and an environmentally dependent, turbulence-based SFE] and two different feedback prescriptions, where only one sub-sample includes early stellar feedback. We use these simulations to assess the impact of different baryonic physics on the H i content of galaxies. We find that the galaxy-wide H i properties agree with each other and with observations. However, differences appear for small-scale properties. The thin H i discs observed in the local universe are only reproduced with a turbulence-dependent SFE and/or early stellar feedback. Furthermore, we find that the morphology of H i discs is particularly sensitive to the different physics models: galaxies simulated with a turbulence-based SFE have discs that are smoother and more rotationally symmetric, compared to those simulated with a constant SFE; galaxies simulated with early stellar feedback have more regular discs than supernova-feedback-only galaxies. We find that the rotational asymmetry of the H i discs depends most strongly on the underlying physics model, making this a promising observable for understanding the physics responsible for shaping the interstellar medium of galaxies.