An
expression meaning to tread very carefully around something, usually used in the sense of avoiding a
topic of conversation, but it can also mean being very cautious in general.
For example:
After the death of Patty's
husband, everyone was walking on eggshells around the subject with her.
After being elected by a very narrow
margin, President Gush (Bore?) was walking on eggshells in creating policy, trying to please everyone.