A historic change is coming to the S&P 500 and MSCI Inc. indexes on Sept. 16, when publicly traded real-estate investment trusts move out of the financial-services sector and into a sector of their own.

“The average investor has no idea what’s going on, but this is a big shift,” says Robert Gordon, who heads Twenty-First Securities, a brokerage firm in New York known for its tax strategies.

The...