Element- and orbital-selective magnetic coherent rotation at the first-order phase transition of a hard uniaxial ferrimagnet

$3d\text{\ensuremath{-}}4f$ intermetalic compounds with heavy rare-earth elements show first-order phase transitions in high magnetic fields due to the competition between the exchange interaction and the magnetocrystalline anisotropy. However, the microscopic picture of the field-induced noncollinear magnetic structures remains elusive. Here we report the direct experimental observation of the coherent stepwise rotation of the $3d$ and $4f$ magnetic moments of the uniaxial hard ferrimagnet ${\mathrm{TmFe}}_{5}{\mathrm{Al}}_{7}$ by using soft x-ray magnetic circular dichroism in pulsed magnetic fields up to 25 T. The element- and shell-selective moments show a transition from the collinear ferrimagnet toward the forced ferromagnetic state via a canted phase, which is explained by a two-sublattice model.