Its not clear to me what you mean when you say you "have" an irrevocable living trust. Are you the beneficiary, or the settlor? Are you the trustee? Without knowing more it will be hard to help.
Also, understand that a bank doesn't "have to" open a particular kind of account for someone. The relationship is private, and banks as legal persons have presumptive freedom of contract. A strange request like a debit card for an irrevocable trust will raise enough flags with most banks for them to just say no.
For persons who are using trusts to plan their own estates, "revocable" are far more common than irrevocable. Irrevocable trusts are generally used with high value estates or special kinds of property. Due to how comparatively less commmon they are, for that reason alone they will increase bank reluctance to cooperate.
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I agree with Attorney Kalamaros' answer. I would simply add that if you are designating the trust as beneficiary, the designation would be, "John Smith, Trustee of the Derek Smith Irrevocable Trust, under agreement dated January 1, 2012." You ARE designating a living person, (the trustee), as the beneficiary. I do not see why the bank should have a problem with this. If they do, you can always move the money to a different bank that is easier to work with.
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