Learn Croatian: Prljav je usprkos tome što ima sapun. - He is dirty despite the fact that he has soap.

In Croatian, prepositions like usprkos cannot directly attach to a clause starting with što. You must introduce a pronoun in the required case first. Here, we use tome (that) to create the set phrase usprkos tome što, meaning despite the fact that (literally: despite that which).

What case is dictated by the preposition usprkos?

It requires the dative case. That is why we use tome, which is the dative form of the demonstrative pronoun to (that).

Is there a common synonym for usprkos that works exactly the same way here?

Yes, unatoč means exactly the same thing and also requires the dative case. You can perfectly well say Prljav je unatoč tome što ima sapun.

Since the verb ima (has) requires an object in the accusative case, why hasn't the noun sapun changed its ending?

It actually is in the accusative case! Because sapun (soap) is a masculine inanimate noun (an object, not a living being), its accusative form looks completely identical to its nominative form.