In assemblies, the geometric frustration of a locally preferred packing motif leads to anomalous behaviours, from self-limiting growth to defects in the ground state(1). Here, we demonstrate that geometric frustration selects the equilibrium morphology of cohesive bundles of chiral filaments, an assembly motif critical to a broad range of biological and synthetic nanomaterials. Frustration of inter-filament spacing leads to optimal shapes of self-twisting bundles that break the symmetries of packing and of the underlying inter-filament forces, paralleling a morphological instability in spherical two-dimensional crystals(2-4). Equilibrium bundle morphology is controlled by a parameter that characterizes the relative costs of filament bending and the straining of cohesive bonds between filaments. This parameter delineates the boundaries between stable, isotropic cylindrical bundles and anisotropic, twisted-tape bundles. We also show how the mechanical and interaction properties of constituent amyloid fibrils may be extracted from the mesoscale dimensions of the anisotropic bundles that they form.