ACR X HAD NMAC AND LTSS FROM MLG DEPARTING MULTIPLE RWY OP PARALLEL RWYS. SYS ERROR.

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ACR X CLRED INTO POS ON RWY 29L AT MSP ON TWR FREQ 126.7. WE WERE TOLD THERE WOULD BE A DELAY IN POS DUE TO CROSS OVER TFC. AN LGT DEPARTED 29L AHEAD OF US. WE WERE CLRED FOR TKOF ON RWY 29L. RWY HDG TO 5000 FT. FO WAS FLYING THE ACFT. TKOF WAS NORMAL; REDUCED THRUST WAS USED. AS WE ROTATED AN MLG CAME INTO MY SIGHT DEPARTING RWY 29R. THE MLG WAS ON ANOTHER TWR FREQ. INITIALLY THE MLG APPEARED TO BE ON A RWY HDG BUT I THEN NOTICED THE START OF A L TURN ACROSS OUR PATH. THE MLG WAS HIGHER THAN US BUT WAS NOT CLBING AS QUICKLY. I TOLD THE TWR CTLR OF THE MLG TFC AND BY THIS TIME FO SAW THE TFC AND LEVELED OUR ACFT OFF TO PASS UNDER THE MLG. I DON'T BELIEVE THE MLG SAW US. HE DID NOT APPEAR TO TAKE ANY EVASIVE ACTION. UPON REALIZING THE CONFLICT THE TWR CTLR ISSUED US AN IMMEDIATE L TURN TO 230 DEG AND A FEW SECONDS LATER A R TURN TO 280 DEG. WE COMPLIED WITH BOTH COMMANDS. WE LOST SIGHT OF THE MLG AS IT PASSED OVER US FROM R TO L. WE PASSED DIRECTLY BENEATH THE MLG BY APPROX 200-300 FT. THIS LGT ACFT WAS NOT EQUIPPED WITH TCASII. WE CONTINUED OUR CLB TO 5000 FT AFTER THE MLG WAS IN SIGHT S OF OUR POS. OUR PROBLEM MAY HAVE BEEN CAUSED BY LACK OF COM BTWN 2 TWR CTLRS ON SEPARATE FREQS. TO PREVENT THIS FROM OCCURRING ACFT SHOULD NOT BE GIVEN TURNS AFTER DEP FROM A RWY THAT CAUSE THEM TO CROSS ANOTHER ACTIVE RWY CTRLINE UNLESS BOTH RWY DEPS ARE TALKING TO THE SAME TWR CTLR. PLT EVASIVE ACTION PROVIDED VERT SEPARATION. ATC TURN COMMANDS PROVIDED NO SEPARATION.