Watermarking should identify language-model output without degrading quality or limiting verification to the model provider. Multilingual deployment makes this harder because morphology, segmentation, and script change where watermark evidence can enter naturally. We introduce LUNA, a linguistically adaptive watermark that combines model-free detection with single-token non-distortion under the standard random-key model. LUNA estimates normalized next-tag entropy from part-of-speech contexts in an external corpus and uses it to set the depth of a non-distortionary binary tournament sampler; the detector reconstructs the same schedule from text, a tokenizer, a tagger, and a secret key. We evaluate six typologically diverse languages and two domains against eight primary baselines. LUNA attains an AUROC of 0.9959 and the lowest mean absolute median perplexity shift of 0.045 across the twelve settings; its 95% bootstrap interval [0.022, 0.073] lies below all baseline intervals. LUNA also records the lowest mean Self-BLEU, Distinct-1, surprisal, and entropy shifts. It is the only method that simultaneously achieves AUROC > 0.99 and an absolute median perplexity shift below 0.1 in a majority of settings, reaching this regime in 9 of the 12 settings while no baseline reaches it in more than 2. Our code is available at: https://github.com/Shinwoo-Park/luna_watermark