What do you say to your husband who has dementia who thinks you are someone else and keeps looking for his wife, who is me and I'm here?

I like CWillie's suggestion, but you're writing that it might not be workable. Still, I would give it a try. I was thinking that perhaps there are some clothes you wear regularly that he recognizes. When he becomes confused, your alter ego can say something to the effect that you're working in another room and she'll get you. Quickly change clothes and return; perhaps he'll recognize you as who you are.

This is really a tough situation though as sometimes there just isn't anything that can be done. My mother had some experiences like that, long before I knew anything about how to address it.

Once when she was staying with me while my father was away, a friend came over and brought some food which Mom loved. She was ecstatic, and excitedly greeted the woman, asking her "are you my daughter?" it was kind of like a sucker punch for me as I hadn't realized the extent to which she sometimes became confused.

And once when my brother came home from out of state to visit, she asked him if he was her son or her husband. At least she was cheerful and excited to see him.

These were years ago and there wasn't as much publicity or information on dealing with these kinds of confusion issues.

Research "Capgras syndrome" and /or "Capgras delusions". If it is Capgras, there are several action you can take to mitigate the symptoms. FYI, Capgras is one of the most challenging symptoms that occur in Dementia. Challenging for both the Care Giver and the Care Receiver. Sorry to provide sad news.

I think it's beyond that. In the beginning I told him I didn't see the other girls and he needed to introduce me to them if he could. So if I act like I see them now I'm sure his reaction would be that of anger and calling me a liar. He get's very hurtful with his words. Which I understand is all part of the disease. I just thought there would be another way to handle this.

This has to be hard! Maybe when he doesn't know you you can say "Ruthann had to pop out to the (store, hair salon, doctor, dentist...) and asked me to come in to help her finish the (dusting, vacuuming, start dinner...). When he sees you "return" you can just give him a big hug and say "I'm here now, what did you want to tell me?"
Try to get to the root of his anger; is he feeling confused by all the "new people", afraid you have left him, does he resent that he has (imagined) sitters because he feels fine?