Sky distribution of the candidate Sgr stream stars. Top: mollweide projection of the histogram, in ICRS coordinates, of the counts obtained from the peaks selected in proper motion (Sect. 2.2). The contours are iso-probability lines of Prob(Sgr|), a score assigned to each source in the final catalogue based on the probability of being a Sgr stream star (see text). In the background, we included a black and white histogram of the input sample for reference. Bottom: histogram of the stars in our final sample, with bins of 1° × 1°, but in the Sgr reference frame (, ). The red (blue) crosses correspond to the peaks of the 1D WT associated with the bright (faint) branch, while the dashed lines represent the resulting polynomials (see Table 1). The Sgr stream appears as a four-tailed structure across the whole sky, for the first time, in a homogeneous astrometric sample.