Chemistry - What happens when you mix alum with soap?

I think you’ve made a so called “aluminium soap”. It is a compound of the form $\ce{(RCOO)3Al}$ where $\ce{RCOO-}$ is a fatty acid chain. The $\ce{Al^3+}$ ions from the alum you handled would have displaced the $\ce{Na+}$ ions in soap to form the thick gel like aluminium version of soap.

This is similar to how $\ce{Mg^2+}$ or $\ce{Ca^2+}$ ions form an insoluble scum (which consists of $\ce{(RCOO)2Mg}$ or $\ce{(RCOO)2Ca}$) when soap is used in hard water.

I assume here that you’re talking about potash alum $\ce{KAl(SO4)2.12H2O}$ as @BenNorris has mentioned in his comment above.

Any other aluminium containing alum will also have the same effect. But chrome alums (containing chromium instead of aluminium) might have a different effect on soap.

These “aluminium soaps” are used as a thickening agent in oils and greases and also in napalm (as Georg has mentioned in his comment above).