Peppermint Grove, a suburb of Perth in Western Australia, has all the trappings you’d expect of one of the wealthiest postal codes in the country: sprawling riverside mansions, exclusive schools and a yacht club.

But lately there’s a new sign that the Perth elite are starting to tip over the line separating the merely rich from the fabulously so. Glance in a real estate agent’s window, and you’ll often see properties advertised with a telltale phrase: "Perfect for a family office.”

Financial hubs such as Dubai, London, New York and Singapore have long dominated the rarefied world of family offices — outfits that typically cater to a single $100 million-plus client with services that can include managing money, taxes, charitable donations and even household help.