We study the Higgs branches of five-dimensional N = 1 rank-zero theories obtained from M-theory on two classes non-toric non-compact Calabi-Yau threefolds: Reid's pagodas, and Laufer's examples. Our approach consists in reducing to IIA with D6-branes and O6-planes, and computing the open-string spectra giving rise to hypermultiplets. Starting with the seven-dimensional worldvolume theories, we switch on T-brane backgrounds to give rise to bound states with angles. We observe that the resulting partially Higgsed 5d theories have discrete gauge groups, from which we readily deduce the geometry of the Higgs branches as orbifolds of quaternionic varieties.
