Nonmonotonic bending-induced changes of fundamental band gaps and quasiparticle energies are observed for realistic nanoscale phosphorene nanosheets. Calculations using stochastic many-body perturbation theory show that even slight curvature causes significant changes in the electronic properties. For small bending radii (\textless4 nm) the band gap changes from direct to indirect. The response of phosphorene to deformation is strongly anisotropic (different for zigzag vs armchair bending) due to an interplay of exchange and correlation effects. Overall, our results show that fundamental band gaps of phosphorene sheets can be manipulated by as much as 0.7 eV depending on the bending direction.