Thanks to her majesty, the royal family has to abide by a set of rules, otherwise called royal protocol. Family girls can't dye their nails with paint, hats aren't shared at most events, attire should always be princess and ... didn't you say anything tattoo?
You can honestly think that they are not allowed to do any tattoos or not? Error!
The queen's niece, Zara Tindall, asked if she would ever allow her children to have a tattoo in the future. The answer showed that this job is not up to her. And to no one else!
"He's got no tattoos!" Zara told her husband. I'd rather have one of my daughters come home with a question than a tattoo. At the very least, it can be removed. I don't know what I would do if I saw them with something like that.
In fact, the princess's granddaughter has even had a question in distant adolescence, proving so with little effort that even royalty can rebel, At any time! Although Zara does not have the same status as Prince Harry, Kate or Meghan, the protocol applies to her.
In short: Harry and William may have some tattoo somewhere, just haven't discovered it yet!