  Frustrated quasidoublets without time-reversal symmetry can host highly
unconventional magnetic structures with continuously distributed order
parameters even in a single-phase crystal. Here, we report the comprehensive
thermodynamic and neutron diffraction investigation on the single crystal of
TmMgGaO$_4$, which entails non-Kramers Tm$^{3+}$ ions arranged on a
geometrically perfect triangular lattice. The crystal electric field (CEF)
randomness caused by the site-mixing disorder of the nonmagnetic Mg$^{2+}$ and
Ga$^{3+}$ ions, merges two lowest-lying CEF singlets of Tm$^{3+}$ into a
ground-state (GS) quasidoublet. Well below $T_c$ $\sim$ 0.7 K, a small fraction
of the antiferromagnetically coupled Tm$^{3+}$ Ising quasidoublets with small
inner gaps condense into two-dimensional (2D) up-up-down magnetic structures
with continuously distributed order parameters, and give rise to the
\emph{columnar} magnetic neutron reflections below $\mu_0H_c$ $\sim$ 2.6 T,
with highly anisotropic correlation lengths, $\xi_{ab}$ $\geq$ 250$a$ in the
triangular plane and $\xi_c$ $<$ $c$/12 between the planes. The remaining
fraction of the Tm$^{3+}$ ions remain nonmagnetic at 0 T and become uniformly
polarized by the applied longitudinal field at low temperatures. We argue that
the similar model can be generally applied to other compounds of non-Kramers
rare-earth ions with correlated GS quasidoublets.
