This could be the story of an American dream. An immigrant family builds a successful business and buys a four-bedroom house in a quiet neighborhood with good schools for their young son. But not all is as it seems on the steep, curving streets of San Diego’s Rancho Penasquitos.

A 45-year-old Chinese woman, Xu Ting, lives in a brown shingle house with a weedy driveway. She has been sued for counterfeiting by eight luxury brands, including Gucci and Louis Vuitton, and owes Chanel Inc. $6.9 million in damages. None of this has stopped her from becoming a legal permanent resident of the United States and achieving a comfortable suburban life.