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When the process underlying DNA substitutions varies across evolutionaryhistory, the standard Markov models underlying standard phylogenetic methodsare mathematically inconsistent. The most prominent example is the general timereversible model (GTR) together with some, but not all, of its submodels. Torectify this deficiency, Lie Markov models have been developed as the class ofmodels that are consistent in the face of a changing process of DNAsubstitutions. Some well-known models in popular use are within this class, butare either overly simplistic (e.g. the Kimura two-parameter model) or overlycomplex (the general Markov model). On a diverse set of biological data sets,we test a hierarchy of Lie Markov models spanning the full range of parameterrichness. Compared against the benchmark of the ever-popular GTR model, we findthat as a whole the Lie Markov models perform remarkably well, with the bestperforming models having eight parameters and the ability to recognise thedistinction between purines and pyrimidines.

A new hierarchy of phylogenetic models consistent with heterogeneous substitution rates