Time phase, as the name suggests, is the unfolding of the process with regard to time: starting vs continuing vs ending. Reality phase, as the name suggests, is the unfolding of the process with regard to reality: unreal (apparent) vs real (realised). Halliday & Matthiessen (2014: 571):
At the deepest level time-phase and reality-phase are the same thing: both are concerned with the stages of becoming. A process is something that emerges out of imagination into reality, like the rising of the sun. Before dawn, the sun shines only in the future, or only in the imagination – as future turns into present, imagination turns into reality.