Discovering high-entropy alloy (HEA) compositions that reliably form a target crystal phase is a high-dimensional inverse design problem that conventional trial-and-error experimentation and forward-only machine learning models cannot efficiently solve. Here we present a ReAct (Reasoning + Acting) LLM agent that autonomously proposes, validates, and iteratively refines HEA compositions by querying a calibrated XGBoost surrogate trained on 4,753 experimental records across four phases (FCC, BCC, BCC+FCC, BCC+IM), achieving 94.66\% accuracy (F1 macro = 0.896). Against Bayesian optimisation (BO) and random search baselines, the full-prompt agent achieves descriptor-space rediscovery rates of 38\%, 18\%, and 38\% for FCC, BCC, and BCC+FCC (Mann--Whitney $p \leq 0.039$), with proposals lying 2.4--22.8$\times$ closer to the experimental phase manifold than random search. An ablation reveals that domain priors shift the agent from landmark-alloy recall toward compositionally diverse exploration -- an uninformed agent scores higher rediscovery by concentrating on literature-dense families, while the full-prompt agent explores underrepresented space (unique ratio 1.0 vs.\ 0.39 for BCC+FCC). These regimes represent distinct criteria: proximity to known literature versus genuine discovery. Spearman analysis confirms agent reasoning is statistically aligned with empirical phase distributions ($ρ= 0.736$, $p = 0.004$ for BCC). This work establishes LLM-guided agentic reasoning as a principled, transparent, and manifold-aware complement to gradient-free optimisation for inverse alloy design.