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Abstract The correlation between two characters is often interpreted as evidence that there exists a significant and biologically important relationship between them. However, Maddison and FitzJohn (in The unsolved challenge to phylogenetic correlation tests for categorical characters. Syst. Biol. 2015;64:127–136) recently pointed out that evidence of correlated evolution between two categorical characters is often spurious, particularly, when the dependent relationship stems from a single replicate deep in time. Here we will show that there may, in fact, be a statistical solution to the problem posed by Maddison and FitzJohn naturally embedded within the expanded model space afforded by the hidden Markov model (HMM) framework. We demonstrate that the problem of single unreplicated evolutionary events manifests itself as rate heterogeneity within our models and that this is the source of the false correlation. Therefore, we argue that this problem is better understood as model misspecification rather than a failure of comparative methods to account for phylogenetic pseudoreplication. We utilize HMMs to develop a multirate independent model which, when implemented, drastically reduces support for correlation. The problem itself extends beyond categorical character evolution, but we believe that the practical solution presented here may lend itself to future extensions in other areas of comparative biology. [Macroevolution; model adequacy; phylogenetic comparative methods; rate heterogeneity].