A Green Card holder is not a foreigner, regardless of their nationality. If anybody attempts to treat you as a foreigner, and discriminates against you because of it, you can file a complaint against them under Fair Housing Law. About the only difference between you and a US citizen is that you can't vote in any elections, and you can't be drafted into the military, otherwise you have the same legal rights and responsibilities as anyone else.
I am very clear on this point because I am not only a licensed real estate professional, I am also a Green Card Brit. Although I have not bought a co-op in NYC, I have purchased many other kinds of real estate in NY, both residential and commercial property, as well I represent clients who have relocated from the UK. You can approach this with your head held high and insist that you are judged by the Co-op Board on the same criteria as anyone else. That is your legal right and there are laws with teeth to protect you in this respect. This is one area of life where America is light years ahead of Europe, and why it is a such a great country to live in.
Good luck with your search.
Can I ask if you want to purchase a Co-op for primary residence, pied a terre or as an investment? Many if not most coops in Manhattan do not like Investors buying in the building, and many of the European buyers buy for that reason, and when they do they have to go with a Condo. If on the other hand you are buying this as a residence for your self, some buildings may give you that right.
I'm looking at your tag and it says you are in Houston, is that Houston England :)?
It is when residency cannot be established that one fails to qualify to finance a purchase. The exception would be a cash sale where the loan qualification process is not involved.
As for the co-op board, national origin is a protected class in accordance with Fair Housing Laws and they cannot be denied because, as you stated, they are "a Green Card Brit".
Drop me a line if you have any further questions or would like to know what is out there in your price range. My number is 917-657-6532 and my email is email@example.com. Cheers!