A system is disclosed for additively manufacturing a composite structure. The system may include a support, a discharge head connected to and moveable by the support, a supply of reinforcement, and a rotary tensioner located between the supply and the discharge head. The rotary tensioner may be configured to impart tension to the reinforcement and to generate a signal indicative of the tension. The system may further include a drive associated with the supply, and a controller in communication with the rotary tensioner and the drive. The controller may be configured to selectively adjust a feed parameter of the drive based on the signal to maintain a desired level of tension.