Symbolic regression (SR) methods have been extensively investigated to explore explicit algebraic Reynolds stress models (EARSM) for turbulence closure of Reynolds-averaged Navier-Stokes (RANS) equations. The deduced EARSM can be readily implemented in existing computational fluid dynamic (CFD) codes and promotes the identification of physically interpretable turbulence models. The existing SR methods, such as genetic programming, sparse regression, or artificial neural networks, require user-defined functional operators, a library of candidates, or complex optimization algorithms. In this work, a novel framework using LLMs to automatically discover algebraic expressions for correcting the RSM is proposed. The direct observation of Reynolds stress and the indirect output of the CFD simulation are both involved in the training process to guarantee data consistency and avoid numerical stiffness. Constraints of functional complexity and convergence are supplementally imposed in the objective function on account of the tremendous flexibility of LLMs. The evolutionary search is employed for global optimization. The proposed method is performed for separated flow over periodic hills at Re = 10,595. The generalizability of the discovered model is verified on a set of 2D turbulent separated flow configurations with different Reynolds numbers and geometries. It is demonstrated that the corrective RANS can improve the prediction for both the Reynolds stress and mean velocity fields. Compared with algebraic models discovered by other works, the discovered model performs better in accuracy and generalization capability. The proposed approach provides a promising paradigm for using LLMs to improve turbulence modeling for a given class of flows.