Turns out that one method for making clear soap involves adding some amount of ethanol and dissolved sugar to the mixture. And some glycerin as well, although why should I do that just because some recipes do and some don't?
Well, attempt number 3 of soapmaking (the first attempt at transparent) went horribly wrong. The soap didn't even saponify properly. That was before I acquired some nice measuring stuff. I now can measure with impunity! Anyway, 3 was a disaster. 4 worked really well for part 1, the saponifying. Part 2, the clarifying, was interesting. With the use of a blender and some Britta filtered methylated spirits (I figured it was the best source of ethanol I was going to get), I went about estimating from some vague online recipes the amounts of ethanol I needed given the amount of Lye. It certainly did turn out translucent, and with my new set of dyes, I went to make bars of translucent soap! Now it turns out that either my calculations were broken, or that glycerin(e)/glycerol actually does something I should know about. Anyway, I now have a bunch of liquid soap that looks like this in close-up:
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I have also acquired some pH test strips which clearly indicate 1-14 using 4 different indicators. Very cool. Also, quite sure my successful soaps are good now and not going to burn anyone's face off.