I asked HSBC about this a few months ago, they have some way of being able to apply for a USA account, but acceptance is not guaranteed. (If they reject your app they very kindly refund the $200 fee.)
The catch was that their bank couldn't accept cheques via mail, but ACH should still work, then to get funds back to your "home" account you send a wire.
I'm in Australia but presumably this would also be possible elsewhere. I don't know anything about the criteria for application, whether they're checking you're solvent, or whether they want to know more detail about what business you're conducting.
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