This article considers the distributed adaptive neuro-fuzzy output feedback control protocol design to solve the output synchronization problem for heterogeneous multiagent systems with nonlinear strict-feedback agent dynamics. The output constraints and actuator saturation are considered simultaneously. First, a distributed high-gain observer is employed to estimate the unmeasured agent state and relax the requirement of the Lipschitz continuity of nonlinear follower dynamics. Second, an asymmetric barrier Lyapunov function with time-varying constraint is presented to deal with both the transient and the steady-state constraints on the output synchronization error. To avoid the 'explosion of complexity,' the dynamic surface control technique is employed to filter the virtual control signal for each follower. To deal with the actuator saturation, a distributed auxiliary dynamical system is designed for each follower. The fuzzy logic system is employed to compensate for the uncertain follower dynamics with guaranteed semiglobal uniformly ultimately boundedness of all closed-loop signals. Finally, a simulation example is conducted to verify the efficacy of the presented adaptive neuro-fuzzy controller design.