Combined neutron and x-ray diffraction experiments demonstrate the formation of a low-temperature minority tetragonal phase in Ba$_{0.76}$K$_{0.24}$Fe$_2$As$_2$ in addition to the majority magnetic, orthorhombic phase. A 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 Ba$_{1-x}$Na$_x$Fe$_2$As$_2$ ($0.24 leq x leq 0.28$), 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.
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