Although the question is quite old I just ran into it. You might still want to (if you haven´t done so yet) look at the article:
A.H. Stone, Inverse limits of compact spaces, General Topology and its Applications 10 Issue 2 (1979) pp 203–211 https://doi.org/10.1016/0016-660X(79)90008-4.
It contains some positive results.
