I don't think people talk too much about the different kinds of "ash" on soap. Maybe it's because people don't realize there are different kinds.
Kevin Dunn has speculated about the different kinds of ash in Chapter 15 of his book
Scientific Soapmaking. I suspect he's also discussed this in some of his lectures to soap makers. (Maybe his lecture outlines are online somewhere?)
He lists sodium carbonate, soap crystals, and sodium hydroxide as three reasonable possibilities. Knowing how fast sodium hydroxide reacts with carbon dioxide in the air, I doubt NaOH "ash" stays as NaOH for long, but it might appear for a short while. (A good candidate for NaOH ash might have been Anna Marie's Andalusian super-lye castile many of us tried to make with a few years ago!)
For long-term "ash" on soap, however, only sodium carbonate (soda ash) and soap crystals make any sense. Here is some of my soap to illustrate the difference --
First photo -- The upright bar has ash; the bar lying flat is "clean." The ash scattered over the top of the upright bar feels exactly like the non-ashy surfaces nearby -- slightly waxy and smooth. If I close my eyes, I can't tell where this ash is just by feeling with my fingertips. It does not easily dissolve with an alcohol or water rinse. If I really scrub on the white parts with a wet cloth, or if I use the bar for bathing, the white bits do wash away after a few minutes of use. IMO, this is soap-crystal ash.
This white ash is only on the top -- the beveled edges, bottom, and sides are clean. I would have done the beveling at the same time the soap bars were cut, 18-24 hours after the soap was made. I'm certain this ash wasn't visible until a few days later however -- I've watched the soap to learn when this ash forms, and it typically takes several days after the bars are cut.
I wish I knew what's different about the unprotected top of the soap that causes this white layer to form. Chemical changes from being exposed to air during saponification? The top stays a little cooler so the soap crystals can form easier? Dunno.
Second photo -- These are a couple bars of my "superlye castile" soap. The white ash on these bars is scattered all over the soap. It feels rough and powdery to me -- I can detect it with my eyes closed. It washes off very easily if the bars are rinsed with water. IMO, this is soda ash.