Tuning controllers for strongly coupled multi-input multi-output (MIMO) industrial processes is hard: decentralized classical auto-tuning ignores loop interaction, and local numerical optimization from natural initializations stalls in the resulting non-convex cost landscape. We ask whether on-premise open-source large language models (LLMs), which keep data on-site and need no plant model, can help. On a single-loop CSTR, classical relay-feedback tuning (IAE 0.106, near the 0.102 optimum) beats an LLM tuner (0.162): for simple loops the LLM adds nothing. The picture inverts on a strongly coupled quadruple-tank with conflicting set-points, scored by a penalized cost J = IAE + lambda*TV(u) that rewards tracking without chattering actuators. There, naive relay tuning (J ~ 28.6) and naive LLM tuning (29.7) are no better than open loop (22.7), and a local optimizer from balanced starts fails in 10/10 runs. A scaffolded open LLM instead reasons about the coupling, proposes the counter-intuitive asymmetric structure, and reaches J ~ 16.9 +/- 0.2 from any start; refining it with a classical optimizer attains the smooth global optimum (J ~ 12.0, 10/10 vs. 0/10), which even applies a non-obvious negative integral correction decentralized tuning cannot. A global optimizer (differential evolution) also reaches this optimum, so the LLM is not the only route; its advantage is sample efficiency and interpretability: a usable controller in 18 evaluations (where the global optimizer is worse than open loop) plus a stated rationale. This edge grows with dimension, reaching ~6x fewer evaluations on a 3x3 plant. The behaviour generalizes across four open models, and on a benign plant the LLM offers no advantage, sharpening the boundary. We contribute a reproducible benchmark delimiting when open LLMs help in control tuning: not as optimizers, but as a sample-efficient, interpretable structural prior.