I have looked around similar questions but none apply directly to my case. For example, this answer is not appropriate for the US and Canada because these countries do not have exit controls. I am a citizen of countries A and B, but I have different names on both, which causes more confusion. I live in Canada with a work visa on passport A, and my passport B is under the ESTA program which I have obtained.
The golden rule of traveling with two passports to me is "enter and leave with the same passport", but I am confused about this particular situation. If I want to visit the United States, what is the correct procedure?
Buy a roundtrip plane ticket using passport A. Then, Canada knows I am leaving the country. When I am at US immigration, I show passport B so that I can enter under the VWP. However, when I leave the US back to Canada, the airline will input that person A is leaving the US, and the US will think I overstayed under B?
Buy a roundtrip plane ticket using passport B. I can enter the US and leave it with no problems. I can then enter my A passport at immigration control in Canada. But since Canada also does not have exit controls, Canada will have seen that person B (that never entered the country) left Canada. Moreover, when I get back to Canada and use passport A, they will also find it strange since in their eyes I had never left Canada in the first place. Would this cause a problem?
Buy two one way tickets. Canada to US under A, then US to Canada under B. Then, I can leave Canada with A (so I closing my entry), then enter and leave US with B (roundtrip with same passport), and enter Canada again with A. It is convoluted, but seems to make sense to me?