When Apple Inc. held its annual hardware showcase last month, its new iPhones got the attention. But the company also announced an accessory that is potentially crucial for its China turnaround: a slim woven strap that lets users wear their handsets like a handbag.

The $59 Crossbody Strap drew curiosity in the US, especially among consumers unfamiliar with the fashion-meets-function trend. But in much of Asia, where Apple faces growing competition from local brands, the idea is hardly new. For years, shoppers have had their pick of colorful, inexpensive cross-body bands in markets and malls across the region.