I commented on an answer, which caused the person who answered to edit his answer accordingly. He added a comment acknowledging my comment. The nature of the edit was such that (1) I didn't have confidence to make the edit directly, and (2) commentary about the edit holds no value to future readers.
The best final state is for both comments to be removed. They should be removed at the same time since neither comment makes sense without the other (now that the question has been edited).
What is the best way to atomically remove the comments?
So far the atomic aspect of this question has proven elusive to a fully satisfactory answer. Since you can't flag your own comment as obsolete, the next best option is deleting your comment and flagging the other. However, this leaves the comments in an inconsistent state (where the remaining comment makes no sense in isolation) until a moderator acts on the flag. Even if you could flag your own comment, since there is a minimum 5 second flag interval and since moderators can't atomically remove comments, there would still a period of inconsistency, albeit typically momentary.
This discussion reminded me that I'd run into a similar situation regarding a comment on a question. I found the related feature request: Support cascading deletion of comments.