Streamer discharges play a central role in electric breakdown of matter in pulsed electric fields, both in nature and in technology. Reliable and fast computations of the minimal model for negative streamers in simple gases such as nitrogen have recently been developed. However, photoionization was not included; it is important in air and poses a major numerical challenge. The authors here introduce a fast and reliable method to include photoionization into our numerical scheme with adaptive grids, and they discuss its importance for negative streamers. In particular, they identify different propagation regimes where photoionization does or does not play a role.