The East Fork. The mighty Virgin River has two main forks: the North and the East. The North Fork is the more famous and accessible, as it carves "The Narrows" and runs through Zion's main canyon. The East Fork of the Virgin, on the other hand, begins near Mount Carmel on Hwy 89 and travels through rugged, roadless backcountry to meet the North Fork near Rockville, Utah. Though the logistics of a Barracks hike are more difficult, the beauty and solitude are worth your trouble.
The East Fork carves its own deep, narrows canyon - Parunaweap Canyon. A close cousin to the Zion Narrows, Parunaweap features the same grandoise river canyon scenery, but not as many miles of it. "The Barracks" specifically refers to the dramatic narrows section of the canyon where the walls are super tall and tight, much like the North Fork Narrows. There are also many excellent side canyons to explore, complete with the bushwhacking and route-finding you'd expect with a little-traveled area.
Lower Parunaweap Canyon, near Rockville, is an off-limits Park Research Area and closed to hikers, so most people cannot access Parunaweap from the "Zion" side. Backpackers interested in exploring The Barracks and Upper Parunaweap must use ingress and egrees points east of Zion, which I detail below. This drainage is associated with the Fat Man's Misery hike in this guide.