This is the result of some research on the structure of American Presidential Elections. Now, this rule is not very well known, but electors in the electoral College are forbidden from casting both their votes for president and vice-president for candidates from the same state. Normally, this would not matter to much. However, there is one year in recent times when this almost came into effect: 2000. Both Bush and Cheney were legally residents of Texas (so this applies in 2004, but that is irrelevant). In OTL, Cheney switched his voting registration to Wyoming, and that was the end of that.
Let's say that people forget about this rule. I'd expect somebody to realize it, but it is concievable that the Republicans miss it until it is to late. So it is Election night in the US. Suddenly, everybody realizes that a vote for Bush means that they cannot vote for Cheney, and vice versa. How do things play out. President Bush and Vice-President Lieberman? President Gore and Vice-President Lieberman? What happens next?