Many natural systems posses, and can transition between, multiple alternative states. For example, a climate ``tipping element'' is a climate component that can transition to an alternative steady state due to an external perturbation such as global warming. Despite the potential impact, existence of alternate states in realistic, complex simulations (e.g. climate models) remain poorly understood. Arguably a reason for this is the lack of applicable methodology that explicitly targets finite yet high-dimensional datasets. In this work we utilize recent progress in computational nonlinear dynamics to formulate a workflow that analyses potentially multi-state simulation data and decides algorithmically what are the alternate states contained within, if any are clearly distinguishable. The framework undergoes an optimization routine that showcases which observables in the data best differentiate the alternate states, and which ones do not differentiate at all, which could be used to guide monitoring and early-warning for multistable components in climate or ecosystems. Finally, once the alternate states have been found, we define an indicator called ``intermingledness''. It quantifies differences and similarities between alternate states, as well as for their basins of attraction (if applicable), across various diagnostic variables. We analyse and present results using three diverse climate datasets: Atlantic ocean circulation, atmospheric midlatitude flow, and habitability of exoplanets. The method is not exclusive to climatic data, but applicable to a variety of cases, including complex networks such as power grids or biological networks. We also provide easy-to-use open source code for applying the workflow to new data.

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