<p>At present, there is an abundance of results showing that the phase-field approach to fracture in elastic brittle materials — when properly accounting for material strength — describes the <i>nucleation</i> of fracture from large pre-existing cracks in a manner that is consistent with the Griffith competition between bulk deformation energy and surface fracture energy. By contrast, results that demonstrate the ability of this approach to describe Griffith fracture <i>propagation</i> are scarce and primarily restricted to Mode I in the setting of infinitesimally small deformations. Aimed at addressing this lacuna, the main objective of this paper is to show that the phase-field approach to fracture describes Mode III fracture propagation in a manner that is indeed consistent with the Griffith energy competition. This is accomplished via direct comparisons between phase-field predictions for fracture propagation in the so–called <i>trousers</i> <i>test</i> and the corresponding results that emerge from the Griffith energy competition. The latter are generated from full-field finite-element solutions that — as a second main contribution of this paper — also serve to bring to light the hitherto unexplored limitations of the classical Rivlin-Thomas-Greensmith formulas that are routinely used to analyze the trousers test.</p>