  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.
