  The Sagittarius stream is one of the best tools that we currently have to
estimate the mass and shape of our Galaxy. However, assigning membership and
obtaining the phase-space distribution of the stars that form the tails is
quite challenging. Our goal is to produce a catalogue of RR Lyrae stars of
Sagittarius and obtain an empiric measurement of the trends along the stream in
sky position, distance and tangential velocities. We generate two initial
samples from the Gaia DR2 RR Lyrae catalogue: one, selecting only the stars
within \pm20deg of the orbital plane of Sagittarius (Strip) and the other, the
result of applying the Pole Count Map (nGC3) algorithm. We then use the
model-independent, deterministic method developed in this work to remove most
of the contamination by detecting and isolating the stream in distance and
proper motions. The output is two empiric catalogues: the Strip sample
(higher-completeness, lower-purity) which contains 11 677 stars, and the nGC3
sample (higher-purity, lower-completeness) with 6 608 stars. We characterise
the changes along the stream in all the available dimensions, the 5 astrometric
ones plus the metallicity, covering more than 2pi rad in the sky and obtain new
estimates for the apocentres and the mean [Fe/H] of the RR Lyrae population.
Also, we show the first map of the two components of the tangential velocity,
thanks to the combination of distances and proper motions. Finally, we detect
the bifurcation in the leading arm and report no significant difference between
the two branches, either in metallicity, kinematics or distance. We provide the
largest sample of RR Lyrae candidates of Sagittarius, which can be used as an
input for a spectroscopic follow-up or as a reference for the new generation of
models of the stream through the interpolators in distance and velocity that we
have constructed.
