The answer is: Milk of Magnesia (MOM).
Yes, the same stuff that relieves constipation.
You apply it to your underarms in a thin coat. My wife paints it on with a paintbrush. I just pour some on my hands and slop it on.
We haven't noticed any problems with odor, even with heavy perspiration, and we've been using nothing but MOM for about three years. It works as well (if not better) than commercial deodorants, and it by far surpasses any of the other natural alternatives. There is simply no need to look elsewhere. The protection is long-lasting and secure. Case closed, end of story.
MOM works far better than baking soda, which is what I previously used for several years. I'm still a fan of baking soda, but you do have to manage it to get it to work right and not burn out your skin.
The one caveat mentioned above involves the inactive ingredients. In some brands of MOM, there are lots of chemical inactive ingredients. The most concerning is sodium hypochlorite (AKA bleach). I go to great lengths to only use MOM that has no other inactive ingredients besides "purified water." No artificial flavors, colors, preservatives and definitely no bleach. Again, these are listed among the inactive ingredients. All brands of MOM have the same active ingredient--magnesium hydroxide. Sometimes it is hard to find the inactive ingredients on the label, but keep looking, they are there (examples are below).
I buy the DG brand, which is found at Dollar General. I believe the Kroger's brand is also good. There are a few other options that I can't remember right now, but not many.
One internet commenter made the point that bleach is most likely used to purify the water, and that the ingredient "purified water" may have just as many traces of bleach as the labels that actually come out and specifically list it. The good news is that there are other way to purify water rather than with bleach, so there is at least a fighting chance that bleach is not part of the process for all of them. Given that generics are often made at the same locations and by the same people as name brands, there is a chance that going out of my way to buy a brand that doesn't list bleach is a fool's errand, but there is at least a ray of hope. I seem to be one of the few who maintains a small bit of skepticism on this. Most are satisfied that the brands that only list the following two ingredients are safe: Magnesium hydroxide (active) and purified water (inactive).
Here is an example of a GOOD brand:
(click for a better view)
The following is definitely what you DON'T want:
(click for a better view)
I can't take credit for this discovery. There are many folks exploring healthy uses of magnesium--both internally as a nutrient and topically to relieve achy muscles and joints and for skin care. This is one of their many discoveries. For more info, check out the Facebook group "Magnesium Advocacy Group."
MOM has all sorts of other topical uses.
Also, check out this article for similar information on MOM as a deodorant.
As news of MOM as a deodorant is beginning to spread, some companies have seized the moment to build MOM-based products. I suppose they have additional essential oils and fragrances for an additional price, but all that seems unnecessary to me. I prefer my $2 bottle of DG brand MOM. I wonder what the cashier thinks of my colon health when I arrive at the checkout counter with a half-dozen bottles of MOM, but so far that has been the only socially awkward moment in using MOM as a deodorant!