It's just known as the Commonwealth (without "realm") . A realm is an individual country ruled by a monarch, not an association of countries. The Commonwealth is a voluntary association of nations most of which were under British rule at some time in the past, plus the UK. Not all countries once ruled by the UK are in the Commonwealth. It is responsible for fostering cultural and economic cooperation.

Hello Ana! The Commonwealth Realm is a group of nations that have or formerly had Queen Elizabeth II as their monarch. This includes the United Kingdom, Australia, Antingua and Barbuda, and the Bahamas.