The closest thing I have found to a shaving, facial, bath soap is Yardley of London Cocoa Butter soap.
https://www.amazon.com/Yardley-Londo...01IADZO1I?th=1
The ingredients are: sodium tallowlate, water, sodium cocoate, glycerine, cocoa butter, fragrance, shea butter, Vitamin E, salt, titanium dioxide, EDTA, iron oxide. Yellow 3, Red 3.
From a shaving standpoint, it is a tallow based soap with some titanium dioxide and iron oxide pigments to stabilize the lather. Cocoa butter and shea butter are good skin conditioners. This is an English milled bar, meaning it is a hard soap, but not triple milled hard. It has a nice smell and and produces a decent later.
It is certainly not as good the better soaps designed specifically for shaving, but if you want an inexpensive soap for bathing as well as shaving, this one will work. You can purchase two bars for $6.