Update: I just tried it out. I had a liquid soap from 20% pure coconut at hand anyway. Up to 7% KCl had next to no effect on the viscosity, and so didn't another 3% of NaCl that I threw in “out of despair” afterwards (initially, some sodium soap precipitated, but it dissolves again over time).
I then tested my “scrap” soap. Unfortunately not much scientific precision here, since I don't have numbers on its composition (it contains coconut, castor, palm, canola, sunflower oils… at unknown concentrations).
Initial fluidness is similar to concentrated lye. On addition of KCl, this liquid soap indeed thickens up. At 8% KCl, it's not unlike liquid honey, and still perfectly clear (except for the air bubbles I stirred in).
My tentative explanation for this inconsistency is the fatty acid profile of coconut oil. For sure I will come back to this when my CP liquid soap (high in lauric, but also oleic/linoleic/ricinoleic) is due, and that time with exact numbers!