Combined neutron and x-ray diffraction experiments demonstrate the formation of a low-temperature minority magnetic tetragonal phase in ${\mathrm{Ba}}_{0.76}{\mathrm{K}}_{0.24}{\mathrm{Fe}}_{2}{\mathrm{As}}_{2}$ in addition to the majority magnetic, orthorhombic phase. The coincident enhancement in the magnetic ($\frac{1}{2} \frac{1}{2}$ 1) peaks shows that this minority phase is of the same type that was observed in ${\mathrm{Ba}}_{1\ensuremath{-}x}{\mathrm{Na}}_{x}{\mathrm{Fe}}_{2}{\mathrm{As}}_{2}$ ($0.24\ensuremath{\le}x\ensuremath{\le}0.28$), in which the magnetic moments reorient along the $c$ axis. This is evidence that the tetragonal magnetic phase is a universal feature of the hole-doped iron-based superconductors. The observations suggest that in this regime the energy levels of the ${C}_{2}$ and ${C}_{4}$ symmetric magnetic phases are very close.