Some people thought he was a homeless drifter with a possible drug problem. Others thought he was a construction worker who floated from job to job. Most people in the Detroit neighborhoods he frequented knew him only as the lanky, easygoing Rodriguez (some claimed his first name was Sixto, others Jesús). He sang songs about longing and the working-class and dreams. He liked to describe himself as a musico-politico. He always, always wore sunglasses, almost like they were armor. One night in the late 1960s, two record executives saw him performing at a bar and came away convinced they had just seen a talent comparable to Bob Dylan — a musical genius waiting to be discovered.

The producers, who had worked with Marvin Gaye and Stevie Wonder, collaborated with Rodriguez on his first album, Cold Fact. Released in 1970, it was an enormous flop. His follow-up, 1971’s Coming from Reality, also failed. No more albums followed. Never much of a social creature, Rodriguez retreated further into obscurity and depression. No one, not even the people who had worked with him at the record labels, stayed in touch. He would make occasional appearances at bars and clubs, the crowds enthusiastic and receptive to his work. Soon, though, he’d disappear again.

And then rumors began to spread that Rodriguez had killed himself onstage at the end of a performance. Some said he had doused himself with gasoline and set himself on fire. Others said he produced a gun and shot himself in the head. Regardless of which story was true, no one knew where Rodriguez was. Meanwhile, in apartheid-ruled South Africa, his songs snowballed in popularity — banned by the government but bootlegged and passed around by fans who were inspired by his anthems of defiance and individuality. Eventually, his records outsold Elvis Presley.

Part of the appeal of Searching for Sugar Man, director Malik Bendjelloul’s engaging, cleverly structured documentary, is that the film is shaped like a mystery. The more we learn about the cultural and political impact of Rodriguez’s music in South Africa — and the more tantalizing snippets of his songs we hear on the soundtrack — the more we want to know about the singer. The filmmakers travel from Cape Town to Detroit to Los Angeles, piecing together the clues of his life. He didn’t leave much to go on, and his mystique undoubtedly helped spur the growing popularity of his music. On one level, Searching for Sugar Man is a testament to how music — or painting or literature or any form of art — can take on a life far greater than its creator intended when it happens to connect with the right people at the right time.

And then, via a random posting on a website and a one-in-a-million coincidence, the filmmakers discover what happened to Rodriguez. For roughly an hour, the bulk of the interviews in the film focus on Rodriguez’s collaborators and the people who had used his songs as a tool for social change. But in its final half hour, the movie turns to Rodriguez’s daughters, now grown, who share their memories of their unassuming father and how he had tried — and failed — to stake his claim on 1970s folk music. Eventually, he set all that aside as something to remember fondly but without lament or regret.

Although it seems to play out in linear fashion, like an unfolding investigation, Searching for Sugar Man toys a bit with chronology near the end, withholding footage shot in 1998 (clearly labeled) to heighten the impact of the end of the movie. The cheat is easy to forgive, though, because it gives this already-amazing story a jaw-dropping, fairy-tale ending. “Sometimes a fantasy is best left alone,” someone says in the movie, arguing that history should be left alone and no one should dig too deeply into Rodriguez’s legend. Searching for Sugar Man, though, argues otherwise.