T-branes are a non-abelian generalization of intersecting branes in which the matrix of normal deformations is nilpotent along some subspace. In this paper we study the geometric remnant of this open string data for six-dimensional F-theory vacua. We show that in the dual M-theory / IIA compactification on a smooth Calabi-Yau threefold X-smth, the geometric remnant of T-brane data translates to periods of the three-form potential valued in the intermediate Jacobian of X-smth. Starting from a smoothing of a singular Calabi-Yau, we show how to track this data in singular limits using the theory of limiting mixed Hodge structures, which in turn directly points to an emergent Hitchin-like system coupled to defects. We argue that the physical data of an F-theory compactification on a singular threefold involves specifying both a geometry as well as the remnant of three-form potential moduli and flux which is localized on the discriminant. We give examples of T-branes in compact F-theory models with heterotic duals, and comment on the extension of our results to four-dimensional vacua.