Actually, in English, Chai does not mean tea, it means a specific flavor of tea. If you don't believe me, try ordering some Earl Grey Chai, see what happens.

If your server is Indian, they'll likely react positively, and get you what you want.

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Sure, but "chai tea" is still redundant. I have never used that term and ordered chai in many places without confusion.

it's redundant at a place that serves chai, but it isn't redundant at a place that does not serve chai, because you're skipping the "what is chai" question from whoever you're querying.