That's how some US tax/banking codes already work, so it's not without precedent. I don't remember exactly what it's called. But allegedly it's a hassle for everyone involved, both banks and customers.
Ah, found it: "... is the Foreign Account Tax Compliance Act (FATCA), which was passed in 2010 and will go into effect in January of 2013. The act requires all foreign banks to identify and report on US citizens with accounts holding more than $50,000 in an effort to clamp down on tax evasion. If banks refuse to comply, they could face a punitive 30 percent withholding tax on all payments from the US."
Ah, found it: "... is the Foreign Account Tax Compliance Act (FATCA), which was passed in 2010 and will go into effect in January of 2013. The act requires all foreign banks to identify and report on US citizens with accounts holding more than $50,000 in an effort to clamp down on tax evasion. If banks refuse to comply, they could face a punitive 30 percent withholding tax on all payments from the US."