FLC OF A B737 INADVERTENTLY TAXIED ACROSS THE ACTIVE RWY DURING TAXI OUT FOR TKOF. THERE WAS NO CONFLICT AND THE TWR CTLR TURNED THEM AROUND AND HEADED DOWN THE CORRECT TXWY.

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THIS WAS THE LAST LEG OF A VERY LONG 4 DAY TRIP. MDW TO EWR. AT THE TIME OF OCCURRENCE WE HAD BLOCKED 27 HRS IN 4 DAYS AND ON ARR IN EWR 29 HRS 19 MINS. THE PREVIOUS DUTY DAY WAS 13 HRS 45 MINS AND ON DAY OF OCCURRENCE OVER 13 HRS AGAIN. WE WERE TIRED; VERY TIRED AND THIS WAS CERTAINLY A MAJOR FACTOR. WE RECEIVED TAXI CLRNC TO TAXI TO RWY 31C VIA TXWY P. BOTH THE CAPT AND MYSELF WERE CONFUSED BY THE 'NEW' SIGNAGE AT MIDWAY. THERE SEEMED TO BE A LOT OF SIGNAGE CLOSE TOGETHER AND IT WAS DIFFICULT TO FIGURE WHICH TXWY WENT WITH WHICH SIGN. I CHKED THE TAXI CHART AND TOLD THE CAPT WE SHOULD BE HDG 180 DEGS ONCE ON TXWY P. ON TXWY P AND TAXIING S WE CROSSED RWY 31R AND BEFORE WE KNEW IT; HAD CROSSED THE ACTIVE RWY 31C. WE MISSED TXWY G THAT WOULD HAVE TAKEN US TO RWY 31C. AS COPLT I WOULD NORMALLY BE DOING TAXI CHKS BUT WAS NOT. I WAS OUTSIDE WITH THE CAPT AND WE BOTH GOT CONFUSED ON THIS TAXI OUT. AFTER XING RWY 31C (THE ACTIVE) GND CTL CALLED OUT OUR ERROR AND TURNED US AROUND. THANKFULLY THERE WAS NO CONFLICT WITH ANOTHER ACFT. I BELIEVE THIS TAXI ERROR OCCURRED FOR THE FOLLOWING REASONS: PLT NOT FAMILIAR WITH MIDWAY. WHEN THERE WAS CONFUSION ABOUT TAXI INSTRUCTIONS THE ACFT SHOULD HAVE BEEN STOPPED UNTIL RESOLVED. FATIGUE. GND CTL MONITORING ACFT MOVEMENT. WE WERE THE ONLY ACFT MOVING ON THE FIELD AT THE TIME. MIDWAY ARPT DESIGN. RWYS CLOSE TOGETHER AND TAXI SIGNS DIFFICULT TO UNDERSTAND.