Time- and temperature-resolved phase formation in the equiatomic NbTiVZr refractory high-entropy alloy has been studied in situ using high-energy synchrotron X-ray diffraction and high-speed video imaging during non-equilibrium solidification. Phase formation is shown to be dependent on the solidification conditions. When the melt is undercooled over 80 K it crystallizes as a bcc single-phase solid solution despite solute partitioning between dendrites and interdendritic regions. When the sample is for some time kept in the semisolid state, two additional bcc phases form in the interdendritic regions. The crystal growth velocity for the NbTiVZr alloy, as estimated from the high-speed videos, shows pronounced sluggish kinetics.