Knockout Round Robin Tournament?

I see a lot of tournaments where each player plays each other once (Zurich for example) but they don't get to play with both pieces (like the current women's 2015 world championship match).

Is there a tournament where each player plays an opponent twice, and then a knockout ruling is applied after tiebreaks?  For example:

6 players
1 vs. 2, 3 vs. 4, 5 vs. 6 / 2 vs. 1, 4 vs. 3, 6 vs. 5

Let's say 4 got 1.5 points like Harika after round 4, game 2. instead of eliminating 3, 3 would still possibly play 1, 2, 5 or 6

These 4 would do a tiebreak the next day.  To keep it simple, say 1 and 5 won.   The results of the first round would be:
1 - 2 points
5 - 2 points
4 - 1.5 points
2 - 1 point
6 - 1 point
3 - .5 point

Round 2 pairings would then be:
1 vs. 5, 4 vs. 2, 6 vs. 3 / 5 vs.1, 2 vs. 4, 3 vs. 6

Let's say 1 and 6 won both to speed things up so I can make my point.  4 and 2 have a tiebreak, 2 winning.  The results would then be:
1 - 4 points
6 - 3 points
2 - 2.5 points
4 - 2.5 points
5 - 2 points
3 - .5 point

Round 3 pairings would be:
1 vs. 6, 2 vs. 5, / 6 vs. 1, 5 vs. 2

Player 3 is now eliminated.  Player 4 would play the next possible pairing.  Let's say 1 and 5 won after a tiebreak. The results would then be:
1 - 5.5 points
6 - 4 points
5 - 3.5 points
2 - 2.5 points

Round 4 pairings would be:
4 vs. 5, 2 vs. 6 / 5 vs.4, 6 vs. 2

Player 1 either has 1st place or plays a tiebreak (similar to Zurich, Anand and Nakamura), rapid (not blitz or armageddon).  This gives a player like 3 a chance to come back after an initial loss.  If they lose again in round 2, then they are out.