When “Once,” the small story of an Irish busker singing earnest songs who falls for an earnest Czech immigrant was released in 2006 it enchanted even the most hard-hearted movie critics. It wasn’t just diehard folk fans who wanted to eat “Once’s” two characters alive. Everyone of every temperament, style and taste loved “Once’s” stripped-down approach to the musical, the lack of grandiose dance numbers and groan-inducing punnery in its songs; even the notorious cranks at the Village Voice called it “one of the greatest musicals of the modern age.” It was a slice-of-life musical, a movie about two down-on-their-luck people writing songs on the spot, harmonizing them and then falling in love through developing them, but—in typical jaded indie cinema verite fashion—they remain too meek and earnest to act on their love.

As for me, on first inspection of Glen Hansard plucking away in some public square in Dublin, I wanted to strangle him; I live in New York, and you can’t walk twenty feet in Williamsburg without bumping into one of Hansard’s bearded, gentle, sensitive, self-important devotees. Once the positively adorable Marketa Inglova—a sort of pudgier, Slavic-accented Debra Winger—sets her sights on Hansard, and the two real-life musicians/lovers start to write duets, I softened a little. But I was still put-off by the film’s cutesy gimmicks—the characters are called “Guy” and “Girl,” the Guy explains his recent break-up by improvising a song on a public bus—not to mention the characters’ endless bouts of self-pitying. And the bummer of an ending seemed clichéd and forced in the same way that a traditional rom-com’s upbeat ending is—there was no reason except self-imposed malaise for Guy and Girl to stay apart.

Ironically, the real-life Hansard and Inglova stayed boyfriend and girlfriend, through the making of “Once” and the unexpected hysteria and fame that formed around it, and through their three years of touring with their band The Swell Season. Directors Nick August-Perna, Chris Dapkins and Carlo Mirabella-Davis, whose “The Swell Season” is a document of that tour, have made a film with ten times the candor and poignancy of “Once.” Unhampered by clumsy fictional devices, the film not only makes us fully grasp Hansard and Inglova’s talent; it makes us root for them to stay together. And the directors get unprecedented access to the usually private matter of a fraying relationship, capturing every wounding argument in deep close-up.

The musical footage is electrifying. Annoying as Hansard’s anguished tics might be, he has undeniably solid pipes—bellowing, eyes-a-flutter, he appears to be summoning every demon in his soul and, ignoring his diaphragm altogether, spewing them out of his throat, with no flat notes whatsoever. Inglova is the shyer, sweeter angel on his shoulder. It’s easy to see why these two won an Oscar and continue to sell out clubs all over the world.

But what makes “The Swell Season” so superior to “Once” is that it doesn’t beckon the audience to love every sheepish grin and humble quip of this couple. Much screen time is devoted to their moments of pettiness and petulance, such as Inglova’s unease with fan photos, which forces Hansard to deal with fans alone. And, in the film’s best scene, Hansard explains to Inglova how his scoffing at the experience of receiving an Oscar upsets his mother, who is truly proud of his achievement. He regrets hurting his mother, he explains, to which Inglova—no stranger to discomfort with fame—quietly erupts. “Why can’t you just be happy?” she barks. “Isn’t this what you wanted?” Hansard, instead of taking this advice—which all of the audience will surely concur with—bristles at her words, calling her insensitive. This is the truest account, since perhaps “Annie Hall,” of how oversensitivity could be the biggest cause of a breakup next to verbal abuse. These two are perhaps too humble, and too humbly talented, to handle not only the pressures of celebrity but each other.

“The Swell Season” also excels for its moments of raw, biting humor. Some of Hansard’s temper tantrums are as charming as they are infantile (“They completely changed the photo! They made me more handsome!” he screams at the mishandling of a promotional poster for “Once.”) And Hansard’s bewildered parents, both delighted and fed-up with their son, are a scream. In one scene, Hansard’s father—who sadly drank himself to death before the film was completed—barks four simple words at his son, summarizing more or less what I felt for stretches of “Once,” and what many people impatient with Hansard’s self-loathing may feel at times during “The Swell Season:” “I’m sick of ya!”