  We use the spectro-photometric information of ~219 million stars from Gaia's
DR3 to calculate synthetic, narrow-band, metallicity-sensitive CaHK magnitudes
that mimic the observations of the Pristine survey, a survey of photometric
metallicities of Milky Way stars that has been mapping more than 6,500 deg^2 of
the northern sky with the CFHT since 2015. These synthetic magnitudes are used
for an absolute re-calibration of the deeper Pristine photometry and, combined
with broadband Gaia information, synthetic and Pristine CaHK magnitudes are
used to estimate photometric metallicities over the whole sky. The resulting
metallicity catalogue is accurate down to [Fe/H]~-3.5 and is particularly
suited for the exploration of the metal-poor Milky Way ([Fe/H]<-1.0). We make
available here the catalogue of synthetic CaHK_syn magnitudes for all stars
with BP/RP information in Gaia DR3, as well as an associated catalogue of more
than ~30 million photometric metallicities for high S/N FGK stars. This paper
further provides the first public DR of the Pristine catalogue in the form of
higher quality recalibrated Pristine CaHK magnitudes and photometric
metallicities for all stars in common with the BP/RP information in Gaia DR3.
We demonstrate that, when available, the much deeper Pristine data greatly
enhances the quality of the derived metallicities, in particular at the faint
end of the catalogue (G_BP>16). Combined, both catalogues include more than 2
million metal-poor star candidates as well as more than 200,000 and ~8,000 very
and extremely metal-poor candidates. Finally, we show that these metallicity
catalogues can be used efficiently, among other applications, for Galactic
archaeology, to hunt for the most metal-poor stars, and to study how the
structure of the Milky Way varies with metallicity, from the flat distribution
of disk stars to the spheroid-shaped metal-poor halo. (Shortened)
