Today we are drowning in natural articles packed with “improvers” in the form of colorants, taste enhancers, long shelf life etc. all overwrapped with packaging.
There is no difference with soaps- ALL are natural... at least on the label.
Truly natural products are usually hand made. The producer gives you the ingredients list and his address, ideally if you know him personally but that can be unpractical. But you should avoid soaps with very bright, unnatural colours, strong /artificial smells, very polished soaps and with ingredients you don’t understand.
You also have to be aware of melt & pour soaps produced from ready made base sold in blocks often containing odd ingredients like commercial soaps, which some people melt, add to it fragrances, herbs or other stuff and sell as hand/home made. Ask such people of detailed ingredients and how they make the soap they’re selling.
The huge majority of commercial soaps contains animal fats (sodium tallowate) of tallow and lard which disqualifies them for vegetarians use and not only. Others use cheap palm or palm kernel oil and “improve” them with synthetic detergents.