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Suppose I have the following scenario:
Leaving Vietnam for USA. Registered with the airline using my Dominican Passport which has a valid US visa. Successfully did exit immigration with my EU passport which I used to successfully enter.
Leaving USA for EU. Registered with the airline using my EU passport so that they know I can go to EU without a visa in my Dominican passport. I also show them my Dominican passport which I used to enter the US so that they can do the immigration exit procedure.
Given that the US is paperless when it comes to entry / exit, how can I confirm that the airline successfully did the paperwork for me?