I think there may be a cultural miscommunication happening here, with differing expectations of what "soap" is.
Growing up in a different African country, there were 2 types of soap available - "bath" soap which is what most westerners picture when they think of a bar of soap, and old fashioned "household" soap. The second type typically comes in a long bar, is soft enough to cut with a regular kitchen knife, and can be used for all sorts of things, including washing hands and body if you can't afford bath soap. In our household, we used a brand called sunlight, and it's primary use was for hand-washing clothing. It made loads of bubbles and was great at stain removal. I don't recall how long it lasted compared to bath soap. It sounds like the folks in Uganda are expecting a similar product.
I have no idea how it is made, or if it contains real soap or a syndet, but I think it is likely to be an industrial process as it appears to be extruded - see attached screen shot. The type we used was usually a uniform dark green colour, but cheaper brands are sometimes a mottled tan. I recall a slightly astringent, clean smell.
To replicate the texture in cold process, I agree with the use of at least some potassium hydroxide. They'll also want to add some coconut oil and sugar or starch to the recipe as well, since the bubbles from lard are smaller than they'll be used to.
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