Billionaire brothers Charles Koch and the late David Koch became known for their financial support of conservative and arts causes. But they, and their two brothers, Bill and Frederick, quietly built another legacy: hundreds of millions of dollars worth of high-end real estate.

The four brothers have amassed some of the world’s most spectacular private homes, including a large waterfront compound on Massachusetts’ Cape Cod once owned by Bunny Mellon, sprawling Palm Beach estates, a replica of an old Western town in Colorado...