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Understanding the behavior of charged complex fluids is crucial for a plethora of important industrial, technological, and medical applications. Using coarse-grained molecular dynamics simulations, here we investigate the properties of a polyelectrolyte solution, with explicit counterions and implicit solvent, that is driven by a steady electric field. By properly tuning the interplay between interparticle electrostatics and the applied electric field, we uncover two nonequilibrium continuous phase transitions as a function of the driving field. The first transition occurs from a homogeneously mixed phase to a macroscopically charge segregated phase, in which the polyelectrolyte solution self-organizes to form two lanes of like-charges, parallel to the applied field. We show that the fundamental underlying factor responsible for the emergence of this charge segregation in the presence of electric field is the excluded volume interactions of the drifting polyelectrolyte chains. As the drive is increased further, a re-entrant transition is observed from a charge segregated phase to a homogeneous phase. The re-entrance is signaled by the decrease in mobility of the monomers and counterions, as the electric field is increased. Furthermore, with multivalent counterions, a counterintuitive regime of negative differential mobility is observed, in which the charges move progressively slower as the driving field is increased. We show that all these features can be consistently explained by an intuitive trapping mechanism that operates between the oppositely moving charges, and present numerical evidence to support our claims. Parameter dependencies and phase diagrams are studied to better understand charge segregation in such driven polyelectrolyte solutions.