Lymphoid tissue is a key reservoir established by HIV-1 during acute infection. It is a site associated with viral production, storage of viral particles in immune complexes, and viral persistence. Although combinations of antiretroviral drugs usually suppress viral replication and reduce viral RNA to undetectable levels in blood, it is unclear whether treatment fully suppresses viral replication in lymphoid tissue reservoirs. Here we show that virus evolution and trafficking between tissue compartments continues in patients with undetectable levels of virus in their bloodstream. We present a spatial and dynamic model of persistent viral replication and spread that indicates why the development of drug resistance is not a foregone conclusion under conditions in which drug concentrations are insufficient to completely block virus replication. These data provide new insights into the evolutionary and infection dynamics of the virus population within the host, revealing that HIV-1 can continue to replicate and replenish the viral reservoir despite potent antiretroviral therapy.