@Todd Ziegler -- I need to contribute info that's contrary to your thoughts.
I apologize!
Yes, it is possible to blend synthetic detergents (syndets) with true lye-based soap (a detergent that can be made by "natural" means).
In the industry, these cleansers are called "combars". Many commercial products are combars; Dove being one example. The advantages of a combar over pure soap or pure syndet --
Soap is less expensive than syndets, so including soap => product that is cheaper to make
Soap doesn't clean or lather well in hard water, syndets aren't affected by hard water
A blend of detergents tends to be milder to the skin than a single detergent
I haven't tried adding syndets to bar soap (yet). I've played around with adding small amounts of syndets (under 5% by weight) to liquid soap and found this works fine. No weirdness except some syndets will thicken liquid soap and some don't. I haven't tried syndets in bar soap yet, but others have. There have been the occasional thread about adding syndets to bar soap. Here are two I know of --
https://www.soapmakingforum.com/threads/sodium-lauryl-sulfoacetate-in-cp.45646/https://www.soapmakingforum.com/threads/hybrid-bars.74838/
edit-- Soap is anionic (negatively charged), and you don't want to mix soap with a cationic surfactant (positively charged). One familiar example of a cationic surfactant is BTMS emulsifier. Every once in awhile people ask about mixing BTMS into soap -- it's not recommended.
Anionic detergents that are compatible with soap include sodium lauryl sulfate (SLS), sodium laureth sulfate (SLeS), sodium lauryl sulfoacetate (SLS), sodium cocoyl isethionate (SCI), and sodium C14-C16 Olefin sulfonate (Bioterge).
Polysorbate 80 (PS80) is non-ionic, and cocamidopropyl betaine (CAPB) is amphoteric. --end edit
If you sell soap, be advised a combar is not considered true soap and you cannot sell it under the minimal regs for true lye-based soap. The labeling for a combar has to conform to the FDA rules for cosmetics.