Abstract Motivation Simulating multiple sequence alignments (MSAs) using probabilistic models of sequence evolution plays an important role in the evaluation of phylogenetic inference tools and is crucial to the development of novel learning-based approaches for phylogenetic reconstruction, for instance, neural networks. These models and the resulting simulated data need to be as realistic as possible to be indicative of the performance of the developed tools on empirical data and to ensure that neural networks trained on simulations perform well on empirical data. Over the years, numerous models of evolution have been published with the goal to represent as faithfully as possible the sequence evolution process and thus simulate empirical-like data. In this study, we simulated DNA and protein MSAs under increasingly complex models of evolution with and without insertion/deletion (indel) events using a state-of-the-art sequence simulator. We assessed their realism by quantifying how accurately supervised learning methods are able to predict whether a given MSA is simulated or empirical. Results Our results show that we can distinguish between empirical and simulated MSAs with high accuracy using two distinct and independently developed classification approaches across all tested models of sequence evolution. Our findings suggest that the current state-of-the-art models fail to accurately replicate several aspects of empirical MSAs, including site-wise rates as well as amino acid and nucleotide composition.