We investigate the failure of fiber bundles containing two independent sources of heterogeneity: structural disorder generated by random fiber misalignment and strength disorder arising from random breaking thresholds. The macroscopic constitutive response is derived by averaging over the joint disorder space and is validated against computer simulations. We show that the two disorder sources cooperate in stabilizing failure: finite misalignment can transform a bundle that is perfectly brittle for a prescribed threshold distribution into a quasi-brittle system exhibiting stable precursor damage. The physical origin of this effect is the response of the surviving bundle to the force released when a fiber fails. When both disorders are finite, the density of critical global elongations vanishes as a square root at the onset of damage. The resulting scarcity of nearly unstable fibers allows early avalanches to arrest under force control. The brittle–quasi-brittle phase boundary shifts systematically toward smaller misalignment with increasing system size and exhibits a square-root finite-size scaling. The results indicate that the finite-area brittle region disappears from the interior of the mixed-disorder plane in the thermodynamic limit, while the single-disorder boundaries remain singular. The avalanche statistics further reveal an effective large-event exponent close to 2.2 near the crossover, distinct from the value 3/2 observed in the single-disorder limits. Increasing misalignment also separates ultimate strength from failure stability: the peak load varies nonmonotonically, whereas the final catastrophic avalanche is progressively reduced. These results demonstrate that geometrical disorder can promote progressive, damage-tolerant failure even when it does not monotonically increase the strength of the bundle.