We experimentally demonstrate high-speed correlated random bit generation in real time using synchronized chaotic lasers commonly driven by a laser with dispersive feedback. The dispersive feedback from a chirped fiber Bragg grating induces frequency-dependent feedback delay and thus no longer causes time-delay signature, and resultantly ensures the signal randomness and security of chaotic laser. Driven by the time-delay signature-free chaotic signal, the two response lasers are routed into chaotic states and establish a synchronization with correlation beyond 0.97 while they maintain a low correlation level with the drive signal. Through quantizing the synchronized laser chaos with a one-bit differential comparator, real-time 2.5-Gb/s correlated random bits with verified randomness are experimentally obtained with a bit error ratio of 0.07. Combining with a robust sampling method, the BER could be further decreased to 1×10−4 corresponding to an effective generation rate of 1.7 Gb/s. Bit error analysis indicates that the bit error ratio between the responses is lower than that between the drive and responses over a wide parameter region due to the synchronization superiority of the responses over the drive