It is possible to have a tournament like this, however I don't believe that it is possible to schedule it so that there are 6 rounds, each with 3 matches played simultaneously. The best solution I am aware of is to play 9 rounds of 2 matches as follows:
Court 1 Court 2 Byes
(M1 F2 v M6 F3) (M2 F1 v M4 F4) (M3 M5 F5 F6)
(M3 F4 v M5 F3) (M4 F6 v M6 F5) (M1 M2 F1 F2)
(M1 F6 v M3 F1) (M2 F5 v M5 F2) (M4 M6 F3 F4)
(M3 F5 v M6 F4) (M4 F3 v M5 F6) (M1 M2 F1 F2)
(M1 F5 v M2 F6) (M5 F1 v M6 F2) (M3 M4 F3 F4)
(M1 F3 v M2 F4) (M3 F2 v M4 F1) (M5 M6 F5 F6)
(M1 F4 v M4 F2) (M2 F3 v M6 F1) (M3 M5 F5 F6)
(M1 F1 v M5 F5) (M2 F2 v M3 F6) (M4 M6 F3 F4)
(M3 F3 v M4 F5) (M5 F4 v M6 F6) (M1 M2 F1 F2)
The repeated same sex opposition pairs are M1M2, M3M4, M5M6, F1F2, F3F4 & F5F6.
Hope that helps.
PS I have moved your message here, since replies are not possible in the comments area.
