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Tell them either issue her a secured credit card or you’ll close all your accounts and go to a different bank. As you found out, trying to be nice doesn’t get you anywhere.

Maybe you can become a citizen and then cosign for her. Unsecured debt to people who might go back to their own country and not pay doesn’t seem to be discrimination to me. I don’t think I could go to most other countries and open a credit card.

Credit cards have gotten tough.
They are no longer issuing cards to people that are not employed.
Tell her to keep applying for cards everywhere she goes.
At gas stations, at stores like kmart, kohl’s, target, sears, jcpenny.
Someone will give in.
Of course, never pay interest and pay in full each month.
I hope you are aware that carrying balances will reduce scores….

The problem is that the bank usually gets away with stuff like this. Yes, it is unfair and discriminatory, but ordinary folks have little recourse.

If your wife has a job, then all it takes is a little time to build up her job history and it should be OK then. If she is not working, then you will have to try at other banks. Or ask her to take a temporary job just to get a few paychecks.