PJ wrote in to say he thinks it's probably a walking iris (often called the Apostle plant). As I noted, they do have a lot in common, from the shape of the leaves to the way the stems arch and bend toward the ground, which is what walking irises do.
But in researching the walking irises, I noticed that the petals on the bloom are different. They are a distinct blue, not violet or lavender, and have a different shape, without the iris throat.
Our best guess so far remains the Peruvian Lily. Here's a closeup of the bloom. Can you help ID this plant?