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bank or emi for new passport in caraibi

Where are you resident firstly as that is more important than passport you hold?
 
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The cheapest option - virtual work visa in UAE (about $400) will get you an UAE residence permit in Dubai and then you can open a personal bank account in UAE without a minimum deposit.
You can also try banks in Andorra or Channel Islands if you have at least $200k to park there.
Another option - register a US LLC with your new passport and open a business bank (EMI) account remotely in USA
 
Congratulations! What is your original passport if you're willing to share? Also is it obvious by name/photo that you may not be from Carribean, e.g.: Asian-looking, Li Tsi name.

I have two passports so I have some experience. My original citizenship is Russia and this is quite clear from my surname (think Ivanoff / Petroff). Banks in Europe understand clearly that the CBI passport is not the only one. Banks in other parts of the world - not so much, they don't ask for the original one (I don't volunteer information if not asked). So may be you should look at banks in the parts of the world which are further away from your country of origin.