When can a base runner tag up, when the ball is touched or the out is recorded?

If a fielder bobbles a fly ball but ends up recording the out, when exactly can a runner tag up? Is it upon first contact with the ball, or is it when he secures it for the out? There was some consternation about that during a softball game. The ump ended up copping out and said he didn't pay attention to when exactly the guy tagged up and let the play stand. So I was curious as to what the actual rule was.