We compute the bulk photovoltaic effect (BPVE) in BiFeO3 using first-principles shift current theory, finding good agreement with experimental results. Furthermore, we reconcile apparently contradictory observations: by examining the contributions of all photovoltaic response tensor components and accounting for the geometry and ferroelectric domain structure of the experimental system, we explain the apparent lack of BPVE response in striped polydomain samples that is at odds with the significant response observed in monodomain samples. We reveal that the domain-wall-driven response in striped polydomain samples is partially mitigated by the BPVE, suggesting that enhanced efficiency could be obtained in materials with cooperative rather than antagonistic interaction between the two mechanisms.