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We use the spectro-photometric information of ~219 million stars from Gaia'sDR3 to calculate synthetic, narrow-band, metallicity-sensitive CaHK magnitudesthat mimic the observations of the Pristine survey, a survey of photometricmetallicities of Milky Way stars that has been mapping more than 6,500 deg^2 ofthe northern sky with the CFHT since 2015. These synthetic magnitudes are usedfor an absolute re-calibration of the deeper Pristine photometry and, combinedwith broadband Gaia information, synthetic and Pristine CaHK magnitudes areused to estimate photometric metallicities over the whole sky. The resultingmetallicity catalogue is accurate down to [Fe/H]~-3.5 and is particularlysuited for the exploration of the metal-poor Milky Way ([Fe/H]<-1.0). We makeavailable here the catalogue of synthetic CaHK_syn magnitudes for all starswith BP/RP information in Gaia DR3, as well as an associated catalogue of morethan ~30 million photometric metallicities for high S/N FGK stars. This paperfurther provides the first public DR of the Pristine catalogue in the form ofhigher quality recalibrated Pristine CaHK magnitudes and photometricmetallicities for all stars in common with the BP/RP information in Gaia DR3.We demonstrate that, when available, the much deeper Pristine data greatlyenhances the quality of the derived metallicities, in particular at the faintend of the catalogue (G_BP>16). Combined, both catalogues include more than 2million metal-poor star candidates as well as more than 200,000 and ~8,000 veryand extremely metal-poor candidates. Finally, we show that these metallicitycatalogues can be used efficiently, among other applications, for Galacticarchaeology, to hunt for the most metal-poor stars, and to study how thestructure of the Milky Way varies with metallicity, from the flat distributionof disk stars to the spheroid-shaped metal-poor halo. (Shortened)

The Pristine survey -- XXIII. Data Release 1 and an all-sky metallicity catalogue based on Gaia DR3 BP/RP spectro-photometry