A system for additively manufacturing a composite structure is disclosed. The system may include a print head configured to discharge a matrix-coated reinforcement, and a support configured to move the print head in multiple dimensions during discharging of the matrix-coated reinforcement. The system may further include at least one cure enhancer located offboard the print head, and a controller in communication with the support and the at least one cure enhancer. The controller may be configured to selectively activate the at least one cure enhancer to expose the matrix-coated reinforcement to a cure energy during discharging of the matrix-coated reinforcement.