I am not sure of the exact format of play within each foursome. If all 4 players compete against each other, then what you ask for is impossible, as there must be pairs of people who compete twice as soon as round 2. However if the format is like mixed doubles, then its possible to do better. In the schedule below all partnerships are different and the only repeated opposition pair is F1 vs F2.
(M5 F4 v M6 F3) (M1 F5 v M3 F6) (M2 F1 v M4 F2)
(M2 F3 v M1 F6) (M6 F1 v M3 F5) (M5 F2 v M4 F4)
(M4 F6 v M3 F4) (M5 F3 v M2 F5) (M1 F1 v M6 F2)
