You can't rule anything out without a scientific process of elimination. Maybe it's tallow, stearic acid, castor, coconut... find the soaps that don't bother you and start comparing ingredients.
Including the pH being too basic. That is the easiest one to test with pH strips. If it is more basic than baking soda then you want to reconsider using that soap right away.
Agree 100%. If PH is off on a soap you'd think the maker would address that. Maybe stick to some known brands for a bit!
Update : Today I decided to use Williams mug soap. I expected to get redness and itching, but I was wrong ! Everything was fine and my skin didn`t get any reaction from this soap. I am happy, at least I can use 2 soaps (the other is Palmolive stick), which works good for me. Maybe some of ingredients in particular soaps gives me bad allergy and they make my psoriasis worse. I am really confused now