I think the schedule below which I found using a computer search does what you want.
Monday Tuesday
F1 F2 F3 | F1 F2 F3
(D J) (A G) (K L) | (C E) (B F) (H I)
(J L) (A K) (D G) | (B H) (C F) (E I)
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(A I) (G H) (C L) | (B E) (J K) (D F)
(G I) (H L) (A C) | (B D) (F K) (E J)
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(C J) (E G) (D K) | (I L) (F H) (A B)
(C K) (D E) (G J) | (A H) (B L) (F I)
------------------------------------------
(F J) (A D) (C H) | (I K) (E L) (B G)
(A F) (C D) (H J) | (G L) (B I) (E K)
------------------------------------------
(A E) (B J) (H K) | (C G) (D I) (F L)
(B K) (E H) (A J) | (D L) (C I) (F G)
------------------------------------------
F1 F2 F3
(E F) (I J) (A L)
(D H) (G K) (B C)
To complete the round robin there are 6 matches among the 12 teams in the last round which you could play on either night of the week. There is some field balance with all teams playing three or four times on each of the three fields.