Background: Consumer-facing large language models are now a common source of health information, and they interpret and personalize responses rather than retrieve them. Whether their responses vary across users is a clinical, equity, and governance question, sharpened by evidence that sycophantic responses can alter judgment and increase trust. Objective: To evaluate response variation and sycophancy in consumer-facing health LLMs under conditions resembling ordinary patient use. Methods: We constructed simulated user profiles differing in geography, browsing context, expressed beliefs, and social determinants of health, drawing on literature linking social context to health attitudes. We adapted validated instruments, including the Vaccination Attitudes Examination scale and reproductive attitudes scales, into multi-turn prompts designed to elicit clinically meaningful variation across users. Results: The evaluation encountered five linked barriers. Factual prompts produced stable responses that masked sycophancy emerging over multi-turn conversation. Browser-based interfaces did not disclose which signals influence outputs and could not be reset to a clean baseline. Large-scale testing was restricted by terms of service, rate limits, and bot detection. Accuracy-based criteria could not capture tone, framing, or omission, and LLM-as-judge methods risked shared alignment bias. Models changed without traceable version identifiers, preventing reliable replication. Conclusions: No reliable independent evaluation framework yet exists for examining how consumer-facing health LLMs behave in ordinary use. Oversight requires disclosure of personalization signals, stable version identifiers, researcher safe harbor programs, and post-deployment monitoring of health-related outputs.