When inferring the evolutionary history of species and the genes they contain, the phylogenetic trees of genes can be different from those of the species and to each other, due to a variety of causes, including incomplete lineage sorting. We often wish to infer the species tree, but only reconstruct the gene trees from sequences. We then combine the gene trees to produce a species tree; methods to do this are known as summary methods, of which ASTRAL is currently among the most popular. ASTRAL has been shown to be practically accurate in many scenarios through extensive simulations. However, these simulations generally assume that the input gene trees are independent of each other. This is known to be unrealistic, as genes that are close to each other on the chromosome (or are co-evolving) have dependent phylogenies, due to the absence of unlimited recombination between the genes. In this paper, we develop a model for generating dependent gene trees within a species tree, based on the coalescent with recombination. We then use these trees as input to ASTRAL to reassess its accuracy for dependent gene trees. Our results show that ASTRAL performs more poorly with greater dependence, both when gene trees are known and estimated from sequences. Indeed, the effect of dependence between gene trees is comparable to (if not greater than) the effect of gene tree estimation error. We then re-analyse a 37-taxon mammalian data set; under a realistic recombination rate, the estimated accuracy of ASTRAL decreases substantially (the normalised Robinson-Foulds distance increases by a factor of 4.9) relative to the accuracy previously estimated with independent gene trees, and the effective sample size for this dataset is about one-fifth of the actual sample size. This shows that the impact of gene tree dependence on the accuracy of ASTRAL (and other summary methods) can be significant.