The real-time unmanned aerial vehicle (UAV) trajectory design of secure integrated sensing and communication (ISAC) is optimized. In particular, the UAV serves both as a downlink transmitter and a radar receiver. The legitimate user (Bob) roams on ground through a series of unknown locations, while the eavesdropper moves following a fixed known trajectory. To maximize the real-time secrecy rate, we propose an extended Kalman filtering (EKF)-based method for tracking and predicting Bob's location at the UAV based on the delay measurements extracted from the sensing echoes. We then formulate a non-convex real-time trajectory design problem and develop an efficient iterative algorithm for finding a near optimal solution. Our numerical results demonstrate that the proposed algorithm is capable of accurately tracking Bob and strikes a compelling legitimate vs. leakage rate trade-off.