Global CTF refinement is very useful for diagnosing the presence of significant beam tilt/trefoil during analysis. It would be useful if this capacity was available on the fly, so that alignment issues could also be detected and corrected on the fly.
I would suggest the following:
Show the data for odd and even terms during refinement, so that errors can be detected. In the format currently in the Global CTF logs would be fine. Maybe only do this once refinement exceeds a certain resolution cutoff (say 4 Å).
Optionally apply higher order corrections after a certain resolution cutoff.
This would be very helpful for exactly what Live is good for - providing rapid feedback during data collection and informing decision making.