*

The transverse-field Ising model on the triangular lattice is expected to host an intermediate Kosterlitz-Thouless (KT) phase through a mapping from spins to a complex order parameter residing on each triangular unit. TmMgGaO₄ is a candidate material to realize such physics due to the non-Kramers nature of the Tm³⁺ ion and two-singlet single-ion ground state. Using inelastic neutron scattering, we confirm this picture by determining the leading parameters of the low-energy effective Hamiltonian. Subsequently, we track the predicted KT phase by inspecting the field and temperature dependence of the ac susceptibility, and spin correlations in both reciprocal space and real space via single-crystal neutron diffraction and magnetic total scattering techniques, respectively. Around 4 K, magnetic pair distribution function analysis provides evidence for the vortex anti-vortex pairs that characterize the proposed Kosterlitz-Thouless phase. Although structural disorder plays a significant role in the field-induced behavior of TmMgGaO₄, the magnetism in zero-field appears relatively free from these effects. TmMgGaO₄ is thus a rare example of a dense spin system for which continuous phase transitions can be investigated and controlled.