This isn’t advisable, but would mixing hand soap with shampoo still create a cleansing formula, or might it instead be counterproductive or harmful? What would it turn into?
Soap is (mostly and usually) soap. Shampoo is a detergent. You’d get a soap with some slightly stronger oil/grease/stain properties.
That’s assuming it doesn’t have the usual additives for bounce, shine, softness, etc. - hard to say what those would do for hand and body washing. Nothing bad, but might give a slimy or greasy after-wash.
It really depends on what kind of soap you’re adding to it. If you’re using Dial, Softsoap, Ivory, etc., then they’re mostly the same as your shampoo (I’m assuming that you’re talking about adding a liquid hand soap to your shampoo). In both cases, the main ingredient is likely to be either sodium lauryl sulfate or ammonium lauryl sulfate, which are the most common detergents used in common cleansers. SLS is more prevalent in soaps, ALS in shampoos. SLS is somewhat more irritating to skin than ALS, and it’s also a bit more drying. So, basically, you’d have a shampoo that dries your hair out slightly more than usual.
They’ll probably differ a bit more in their secondary and tertiary ingredients, but you wouldn’t add much from the soap - mostly, the effect would be that the bounce-giving, sheen-giving, anti-dandruff ingredients in your shampoo would be watered down. You might make your shampoo antibacterial, if you used a soap that had antibacterial ingredients in it, but that would be about it. I don’t believe you’d have a chemical reaction from mixing the soap and the shampoo - at the temperatures you’d be exposing the soap/shampoo to, I don’t believe that either is chemically active.
Body wash ftw.
See
Comparatively Speaking: Shampoo vs. Body Wash Formulation-
Per article body wash has detergent base not a soap base.