Visual attention can be divided over multiple objects or locations. However, there is no single theoretical framework within which the effects of dividing attention can be interpreted. In order to develop such a model, here we manipulated the stage of visual processing at which attention was divided, while simultaneously probing the costs of dividing attention on two dimensions. We show that dividing attention incurs dissociable time and precision costs, which depend on whether attention is divided during monitoring or during access. Dividing attention during monitoring resulted in progressively delayed access to attended locations as additional locations were monitored, as well as a one-off precision cost. When dividing attention during access, time costs were systematically lower at one of the accessed locations than at the other, indicating that divided attention during access, in fact, involves rapid sequential allocation of undivided attention. We propose a model in which divided attention is understood as the simultaneous parallel preparation and subsequent sequential execution of multiple shifts of undivided attention. This interpretation has the potential to bring together diverse findings from both the divided-attention and saccade preparation literature and provides a framework within which to integrate the broad spectrum of divided-attention methodologies.