As a matter of policy supported by a wealth of arguments, many banks say "No U.S. TIN, no account."
If your bank is not going to require a U.S. TIN then you should at least require a TIN assigned by the individual's country of origin as a condition of opening the account. The CTR and the 1042S expect you to have one...
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In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.