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There are a couple of factors involved.
1) Most (but not all) of the UV radiation from the Sun is blocked by the atmosphere.
2) The amount of visible light given off through fluorescence is less than the amount of UV light that's shining on it.
So when you have sunlight shining on your tattoo, the amount of re-emitted light is simply swamped by the amount of ambient light.
If you put your tattooed person into a dark room with a small hole to let some sunlight in and then put a filter over the hole that blocked visible light but allowed UV to pass, you'd be able to see the tattoo glow.
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