Upon infection by HIV, a host cell proceeds through a critical choice between replication and latency. Viral replication causes host cell death, while latent infections can persist for decades and represent the most significant obstacle to eliminating HIV from a patient. This fate decision depends on both the chromatin environment at the viral integration site, and the availability of transcription factors within the host cell. We have previously shown that, for some integration positions, HIV promoter activity can result in a bifurcating expression pattern in an infected clonal population, where some cells have high promoter activity while others have low activity. Here we explore how the host transcription factor NF-κB and the viral transactivator of transcription Tat affect expression from the HIV promoter at fixed integration positions. Computational modeling predicts that clonal populations will transition between specific expression profiles if the concentration of either of these factors is systematically varied. We experimentally test our prediction by inducibly expressing Tat or NF-κB in our model system, and directly demonstrate that a clonal population with initially low promoter activity can transition to a bifurcating state. We further show that at this new transcription factor steady-state, stochastic fluctuations drive switching between low and high promoter activity. By quantitatively understanding how transcription factors affect the HIV fate decision, we hope to improve rational design of therapeutic strategies to counteract viral latency.