Fly Ball Rules: Runner Advancement After Dropped Catch

Question
if a runner is on base and a fly ball is hit, the ouutfielder appears to catch the ball and then bobbles it and then catches it. if the runner is tagged up on the initial touch can they advance without retagging.  What if the runner is off the bag on the initial touch do they have to go back and tag because of a caught ball or can they just advance like nothing happened?

Answer
On a caught fly ball, the runner is not allowed to leave the base until the initial touch of the ball.  If they are off the base, they must tag up before advancing.  

There is no automatic out when the rule in violated.  The defense must make a proper appeal.

Tom