For someone who has two non-US passports, one eligible for the Visa Waiver Program (European) and one not eligible for the Visa Waiver Program (Israeli) but which does hold a 10-year, multiple entry B-1/B-2, can they alternate the passports they enter the US on? From what I've read, it seems that "visa runs" either don't work (i.e. short travel to Canada/Mexico is counted as part of the stay in the US) or are frowned upon by the CBP (who may not allow an additional entry after a short stay outside the US). But if the US entries were spaced out between the two passports, could that help?
In this case, the person is married to a US citizen (expat) and thus could apply for a green card, but is not interested in becoming a US person since there are all sorts of tax and foreign account implications. But they are interested in spending a year or two traveling around the US.
Can I have multiple valid ESTAs in different passports? addresses a similar question but for two ESTA-eligible passports.