  The possibility of identifying co-natal stars that have dispersed into the
Galactic disc based on chemistry only is called strong chemical tagging. Its
feasibility has been debated for a long time, with the promise of
reconstructing the detailed star-formation history of a large fraction of stars
in the Galactic disc.
  We investigate the feasibility of strong chemical tagging using known member
stars of open clusters.
  We analysed the largest sample of cluster members that have been
homogeneously characterised with high-resolution differential abundances for 16
different elements. We also investigated the possibility of finding the known
clusters in the APOGEE DR16 red clump sample with 18 chemical species. For both
purposes, we used a clustering algorithm and an unsupervised dimensionality
reduction technique to blindly search for groups of stars in chemical space.
  Even if the internal coherence of the stellar abundances in the same cluster
is high, typically 0.03 dex, the overlap in the chemical signatures of the
clusters is large. In the sample with the highest precision and no field stars,
we only recover 9 out of the 31 analysed clusters at a 40% threshold of
homogeneity and precision. This ratio slightly increases when we only use
clusters with 7 or more members. In the APOGEE sample, field stars are present
along with four populated clusters. In this case, only one of the open clusters
was moderately recovered.
  In our best-case scenario, more than 70% of the groups of stars are in fact
statistical groups that contain stars belonging to different real clusters.
This indicates that the chances of recovering the majority of birth clusters
dissolved in the field are slim, even with the most advanced clustering
techniques. We show that different stellar birth sites can have overlapping
chemical signatures [abridged]
