Human immunodeficiency virus (HIV)-1 exhibits remarkable genetic diversity. For this reason, an effective HIV-1 vaccine must elicit antibodies that can neutralize many variants of the virus. While broadly neutralizing antibodies (bnAbs) have been isolated from HIV-1 infected individuals, a general understanding of the virus-antibody coevolutionary processes that lead to their development remains incomplete. We performed a quantitative study of HIV-1 evolution in two individuals who developed bnAbs. We observed strong selection early in infection for mutations affecting HIV-1 envelope glycosylation and escape from autologous strain-specific antibodies, followed by weaker selection for bnAb resistance later in infection. To confirm our findings, we analyzed data from rhesus macaques infected with viruses derived from the same two individuals. We inferred remarkably similar fitness effects of HIV-1 mutations in humans and macaques. Moreover, we observed a striking pattern of rapid HIV-1 evolution, consistent in both humans and macaques, that precedes the development of bnAbs. Our work highlights strong parallels between infection in rhesus macaques and humans, and it reveals a quantitative evolutionary signature of bnAb development.