  Chemical tagging seeks to identify unique star formation sites from
present-day stellar abundances. Previous techniques have treated each abundance
dimension as being statistically independent, despite theoretical expectations
that many elements can be produced by more than one nucleosynthetic process. In
this work we introduce a data-driven model of nucleosynthesis where a set of
latent factors (e.g., nucleosynthetic yields) contribute to all stars with
different scores, and clustering (e.g., chemical tagging) is modelled by a
mixture of multivariate Gaussians in a lower-dimensional latent space. We use
an exact method to simultaneously estimate the factor scores for each star, the
partial assignment of each star to each cluster, and the latent factors common
to all stars, even in the presence of missing data entries. We use an
information-theoretic Bayesian principle to estimate the number of latent
factors and clusters. Using the second Galah data release we find that six
latent factors are preferred to explain N = 2,566 stars with 17 chemical
abundances. We identify the rapid- and slow-neutron capture processes, as well
as latent factors consistent with Fe-peak and \alpha-element production, and
another where K and Zn dominate. When we consider N ~ 160,000 stars with
missing abundances we find another 7 factors, as well as 16 components in
latent space. Despite these components showing separation in chemistry that is
explained through different yield contributions, none show significant
structure in their positions or motions. We argue that more data, and joint
priors on cluster membership that are constrained by dynamical models, are
necessary to realise chemical tagging at a galactic-scale. We release software
that allows for model parameters to be optimised in seconds given a fixed
number of latent factors, components, and $10^7$ abundance measurements.
