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We investigate the magnetic properties of $S=1$antiferromagnetic diamond-lattice, $\mathrm{Ni}{X}_2{\left(\mathrm{pyrimidine}\right)}_2$ $(X=\mathrm{Cl}, \mathrm{Br})$, hosting a single-ion anisotropy (SIA) orientation which alternates between neighboring sites. Through neutron diffraction measurements of the $X=\mathrm{Cl}$compound, the ordered state spins are found to align collinearly along a pseudo-easy axis, a unique direction created by the intersection of two easy planes. Similarities in the magnetization, exhibiting spin-flop transitions, and the magnetic susceptibility in the two compounds imply that the same magnetic structure and a pseudo-easy axis is also present for $X=\mathrm{Br}$. We estimate the Hamiltonian parameters by combining analytical calculations and Monte Carlo (MC) simulations of the spin-flop and saturation field. The MC simulations also reveal that the spin-flop transition occurs when the applied field is parallel to the pseudo-easy axis. Contrary to conventional easy-axis systems, there exist field directions perpendicular to the pseudo-easy axis for which the magnetic saturation is approached asymptotically and no symmetry-breaking phase transition is observed at finite fields. Published by the American Physical Society2024