This paper investigates the event-triggered consensus control for a class of multi-agent systems (MASs) with input saturations and full-state constraints. First, the mean-value theorem is invoked to transform the structure of the input saturations, and the remained compound nonlinear functions can be approximated by neural networks (NNs). Then, based on the adaptive back-stepping control technique and the barrier Lyapunov function (BLF) method, an adaptive fuzzy consensus controller is designed to ensure that the full-state constraints are not violated. Furthermore, we propose a generalized relative threshold that considers neighbouring controller errors to significantly reduce computational burden. The effectiveness of the proposed control scheme is illustrated by a numerical example.