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Luminous red galaxies (LRGs) are one of the key tracers of the large-scale structure of the Universe used by galaxy surveys. Hence, it is important to make accurate predictions for their properties and clustering, including the errors on these statistics. Here, we describe a novel technique which uses the semi-analytical model of galaxy formation {\sc Galform}, embedded in the high-resolution $N$-body Planck-Millennium simulation, to populate a thousand halo catalogues generated using the Parallel-PM $N$-body {\sc glam} code. Our hybrid scheme allows us to make clustering predictions on scales that cannot be modelled in the original $N$-body simulation. LRGs are selected in the redshift range $z=0.6-1$ from the {\sc Galform} output using similar colour-magnitude cuts in the $r$, $z$ and $W1$ bands to those that will be applied in the Dark Energy Spectroscopic Instrument (DESI) survey. We find that the LRG-halo connection is non-trivial, leading to the prediction of a non-standard halo occupation distribution; in particular, the occupation of central galaxies does not reach unity for the most massive haloes, and drops with increasing mass. The {\sc glam} catalogues reproduce the abundance and clustering of the LRGs predicted by {\sc Galform}, and show good agreement with recent measurements of the clustering of DESI-like LRGs using photometric redshifts. We use the \glam{} mocks to compute the covariance matrices for the two-point correlation function and power spectrum of the LRGs and their background dark matter density field, revealing important differences. We also make predictions for the linear-growth rate and the baryon acoustic oscillations distances at $z=0.6$, $0.74$ and $0.93$. All DESI-like LRG catalogues are made publicly available.