The main result of Condensation and Large Cardinals is the proof that Local Club Condensation can be obtained over any model of GCH by cofinality-preserving forcing while preserving an omega-superstrong cardinal and hence Local Club Condensation is consistent with the existence of an omega-superstrong cardinal. As Local Club Condensation is preserved in initial segments of the set theoretic universe, this shows Local Club Condensation to be consistent with most of the usual large cardinal axioms.

The paper may be found here. We used its technique of proof to verify a stronger result about Local Club Condensation and Acceptability here.

The first proof of the main result of Condensation and Large Cardinals, using a somewhat different proof strategy, can be found in my thesis here, but you're strongly advised to read the simplified and corrected version here instead.