We report the experimental discovery of "electrorheological (ER) complex plasmas," where the control of the interparticle interaction by an externally applied electric field is due to distortion of the Debye spheres that surround microparticles (dust) in a plasma. We show that interactions in ER plasmas under weak ac fields are mathematically equivalent to those in conventional ER fluids. Microgravity experiments, as well as molecular dynamics simulations, show a phase transition from an isotropic to an anisotropic (string) plasma state as the electric field is increased.