Definition. A Proof Theory is complete if whenever (truth/valid) then (proof).

Reminder from Logic: Proof Theory, pronounced “proves” Semantics, pronounced “entails” / “valid” / “semantic entailment”

I am still confused at how these two are related.

If a proof system is unsound and complete, then all the semantically valid inferences can be proven, but in addition it proves some sequents that are not actually valid.