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The Dark Energy Spectroscopic Instrument (DESI) Bright Galaxy Survey (BGS) will be a survey of bright, low-redshift galaxies, which is planned to cover an area of |${\sim } 14\, 000$| square degrees in three passes. Each pass will cover the survey area with ∼2000 pointings, each of area ∼8 square degrees. The BGS is currently proposed to consist of a bright high priority sample to an r -band magnitude limit r  ∼ 19.5, with a fainter low priority sample to r  ∼ 20. The geometry of the DESI fibre positioners in the focal plane of the telescope affects the completeness of the survey and has a non-trivial impact on clustering measurements. Using a BGS mock catalogue, we show that completeness due to fibre assignment primarily depends on the surface density of galaxies. Completeness is high (>95 per cent) in low-density regions, but very low (<10 per cent) in the centre of massive clusters. We apply the pair inverse probability (PIP) weighting correction to clustering measurements from a BGS mock which has been through the fibre assignment algorithm. This method is only unbiased if it is possible to observe every galaxy pair. To facilitate this, we randomly promote a small fraction of the fainter sample to be high priority, and dither the set of tile positions by a small angle. We show that inverse pair weighting combined with angular upweighting provides an unbiased correction to galaxy clustering measurements for the complete three pass survey, and also after one pass, which is highly incomplete. [ABSTRACT FROM AUTHOR]