We investigate how magnetic fields affect the quadrupolar domains in the $5d$ transition-metal oxide ${\mathrm{Ba}}_2{\mathrm{MgReO}}_6$ (BMRO), which crystallizes in the double-perovskite structure. At $T_{\mathrm{q}}=33$ K, BMRO undergoes a quadrupolar-order transition, resulting in a cubic-to-tetragonal structural change, followed by a magnetic transition to a canted antiferromagnetic phase at $T_{\mathrm{m}}=18$ K. These transitions cause the formation of three types of tetragonal domains below $T_{\mathrm{q}}$ and four types of magnetic domains for each tetragonal domain below $T_{\mathrm{m}}$. One tetragonal domain is preferred when the magnetic field is applied along the cubic [1 1 0] direction, and its volume fraction exceeds 80% at 10 K. The magnetic-field direction has a significant impact on domain selection. It is revealed that a magnetic field induces not only magnetic but also structural domain switching via unique spin–lattice coupling in the spin–orbit-entangled quadrupolar-order state in BMRO.

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• Multipole order
• Mott insulators
• X-ray diffraction