I don't think your reaction to antiperspirants is an allergy, it's just that they irritate your skin. My understanding is that you can be allergic only to proteins, not non-organic salts.
That being said, antiperspirants also irritate my skin something fierce. Standard deodorants do nothing but add a little perfume to however I smell.
Several years ago (more than ten), I switched to those "crystal" deodorants. It's a big hunk of what looks like rock salt - you wet it and smear it around your underarms. A single crystal lasts about two years of doing this every day (as long as you don't drop it, 'cause they shatter). They reportedly work by making your skin inhospitable to the bacteria that cause the odor. They work very well for me, but I have to re-apply if I'm going out in the evening. They even completely solved an odor problem I was having with a medical boot a few years ago. After four or five days, it started to stink, so I used the crystal, and went the rest of the six weeks with no odor at all.
The only problem is that stores generally sell them in the "health food" section. I generally get my wife to buy them for me, but if I need to I'll try to get in and out of that section of the store before anyone sees me.