Modelling fluid turbulence using a "skeleton" of coherent structures has traditionally progressed by focusing on a few canonical laboratory experiments such as pipe flow and Taylor-Couette flow. We here consider the stratified inclined duct, a sustained shear flow whose density stratification allows for the exploration of a wealth of new coherent and intermittent states at significantly higher Reynolds numbers than in unstratified flows. We automatically identify the underlying turbulent skeleton of this experiment with a data-driven method combining dimensionality reduction and unsupervised clustering of shadowgraph visualisations. We demonstrate the existence of multiple types of turbulence across parameter space and intermittent cycling between them, revealing distinct transition pathways. With a cluster-based network model of intermittency we uncover patterns in the transition probabilities and residence times under increasing levels of turbulent dissipation. Our method and results pave the way for new reduced-order models of multi-physics turbulence.