Chronic infections of the human immunodeficiency virus (HIV) create a very complex coevolutionary process, where the virus tries to escape the continuously adapting host immune system. Quantitative details of this process are largely unknown and could help in disease treatment and vaccine development. Here we study a longitudinal dataset of ten HIV-infected people, where both the B-cell receptors and the virus are deeply sequenced. We focus on simple measures of turnover, which quantify how much the composition of the viral strains and the immune repertoire change between time points. At the single-patient level, the viral–host turnover rates do not show any statistically significant correlation, however, they correlate if one increases the amount of statistics by aggregating the information across patients. We identify an anti-correlation: large changes in the viral pool composition come with small changes in the B-cell receptor repertoire. This result seems to contradict the naïve expectation that when the virus mutates quickly, the immune repertoire needs to change to keep up. However, a simple model of antagonistically evolving populations can explain this signal. If it is sampled at intervals comparable with the sweep time, one population has had time to sweep while the second cannot start a counter-sweep, leading to the observed anti-correlation. This article is part of the theme issue ‘Interdisciplinary approaches to predicting evolutionary biology’.