For Prince Harry, the issue isn't that he will suddenly end up paying U.S. income tax, but rather that Markle's American citizenship could open up the secretive finances of the royal family to outside scrutiny. If she remains a U.S. citizen, Markle will have to file her taxes to the IRS every year. And if she has more than $300,000 in assets at any point during the tax year — a likely scenario, given her successful acting career and her future husband — she will be expected to annually file a document called Form 8938 that will reveal the detail of these assets, which could include foreign trusts.
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You can bet that the IRS would be very protective of the secrecy of this return if it ever came to be filed. Unauthorized disclosure would almost certainly get you fired and quite possibly subject you to criminal prosecution. She also could choose to file separately, which might keep a lot of her husband's assets off the return.