The Tiered Radio Extragalactic Continuum (T-RECS) simulation II: HI emission and continuum-HI cross-correlation

In this paper we extend the Tiered Radio Extragalactic Continuum Simulation (T-RECS) to include HI emission. The HI T-RECS model is based on the most recent HI mass function estimates, combined with prescriptions to convert HI mass to total integrated HI flux. It further models source size, morphology and kinematics, including rotational velocity and HI line width. The continuum T-RECS model is updated to improve the agreement with deeper number counts available at 150\,MHz. The model for star-forming galaxies (SFGs) is also modified according to the most recent indications of a star formation rate (SFR)--radio luminosity relation, which depends primarily on stellar mass rather than redshift. We further introduce prescriptions to associate an HI mass to the T-RECS radio continuum SFG and Active Galactic Nuclei (AGN) populations. This gives us a way to meaningfully associate counterparts between HI and continuum catalogues, thus building HI $\times$ continuum simulated observations. Clustering properties of the sources in both HI and continuum are reproduced by associating the galaxies to dark matter haloes of a cosmological simulation. We deliver a set of mock catalogues, as well as the code to produce them, which can be used for simulating observations and predicting results from radio surveys with existing and forthcoming radio facilities, such as the Square Kilometre Array (SKA)