Obstructin- With a Twist

[BretMan]

An odd question was posed to me at a coach's rule clinic today. Thought that I would throw this one out there for comments.

Here is the play:

Runner on second base. Batter hits a clean single to the outfield. As runner goes for third, she is obstructed by F6.

Umpire calls and signals "obstruction" and judges to protect runner to home.

The play at the plate is a close one. The catcher misses the initial tag attempt. As the runner slides, she misses the plate and goes past it.

Catcher tags runner on her attempt to return and touch the plate, and does indeed tag her before she gets there.

Has the runner, by sliding past home plate, effectively "overrun" her protection from being put out?

My inclination was that she had and would be called out on the tag. That is the answer that I gave the coach, but also told him I would consult my expert references- that's you guys!- for further clarification.

Any Case Plays, interpretations, stated rulings to cover this?