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Offshore company + bank account without passport? Can I?

Without passport you won't get anything. Maybe you can find a company setup in the USA where you don't have to provide any passport at the beginning but sooner or later you will have to provide it.

The exception is if you are goind the darkroute.
 
absolutely can not, even you use nominee for hidden UBO information.
The bank will want to know the real owner of the company and that will reveal your identity. There are people doing tricks but I don't believe this to be a valid route. You will have to trust a 3rd party all your banking details and have him sign and have access to the bank account. Who want that?o_O