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 five latent factors are preferred to explain $N =$ 1,072 stars with 18 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 only K and Zn dominate. When we consider $N \sim$ 100,000 stars with missing abundances we find another seven 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 accompanying software that scales well with the available data, allowing for model parameters to be optimised in seconds given a fixed number of latent factors, components, and $\sim10^7$ abundance measurements.

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