Kiwibank communications manager Bruce Thompson said the bank did not use facial recognition software but did keep video footage in case of an incident.
"We need to see our customer's faces clearly in case we are robbed - we can give the police clear footage of the people involved," Mr Thompson said.
The rules are clearly displayed within the bank and included helmets, glasses, hats and other attire that could obscure faces.
"Occasionally people get upset, especially if they have prescription sunglasses or for another reason don't feel they should be inconvenienced. Some feel affronted.
"But the rules help protect our staff and act as a deterrent against a possible robbery. We don't use facial recognition software but we do have clear video footage which can be used to identify people if needed."
Mr Himmel's question follows a similar incident earlier this year when a woman objected to being asked to remove her hat.