A CE750 CREW TOLD TO CROSS RWY 4R AT MDW IS THEN TOLD TO HOLD SHORT AFTER ALREADY STARTING ACROSS. B737 ON FINAL GOES AROUND.

Narrative

GND CTL CLRED US TO CROSS RWY 4R AT TXWY F AND TO GIVE WAY TO ACR ON THE OTHER SIDE OF THE RWY WHILE JOINING TXWY Y TO THE RAMP. STREAMS OF ACFT WERE ARRIVING AND DEPARTING RWY 4R. WE CONFIRMED THE CLRNC AND ACKNOWLEDGED VISUALLY THE ACFT ON AN APPROX 1 MI FINAL. AFTER XING THE RWY HOLD LINE AND TAXIING ONTO THE RWY THE CTLR REACTIVELY INSTRUCTED US TO STOP. AT THIS POINT A PORTION OF THE NOSE AND ACFT WERE ON THE RWY. WE CHKED THE ACFT ON FINAL AND AFTER BRIEFLY SLOWING EXPEDITED ACROSS THE RWY AND ADVISED ATC OF OUR CLRING. THE ACFT ON FINAL WENT AROUND. WE CLRED THE RWY AND CONFIRMED FURTHER TAXI INSTRUCTIONS WITH ATC. WE THEN JOINED TXWY Y AFTER ACR AND PROCEEDED TO THE RAMP WITHOUT INCIDENT. AFTER SHUTDOWN WE REVIEWED THE EVENTS AND AGREED WITH THE CHOSEN COURSE OF ACTION. I THEN SPOKE WITH THE CREW OF THE ACFT THAT WAS ON FINAL AND EXPLAINED THE SITUATION. THEY EXPLAINED THAT THEY EXPECTED TO GO AROUND BASED ON THE TFC SITUATION AND WHAT WAS HAPPENING. ALTHOUGH THE CTLRS WERE DOING A GREAT JOB; WE FELT THEY WERE MORE CONCERNED WITH MAINTAINING THE ACR'S TEMPO AND SCHEDULE. LATER WHILE HOLDING SHORT FOR TKOF AT RWY 31C THE ACR QUICKLY CALLED ATC AS TO GO IN FRONT OF US. ALSO ON THE RADIO ANOTHER ACR WAS ADVISED TO TAXI INTO POS AND HOLD BUT NOT STOP AND BE READY TO GO. IF WE HADN'T BEEN INSTRUCTED TO STOP AFTER BEING CLRED TO CROSS THE RWY; WE WOULD HAVE CLRED THE RWY IN TIME FOR THE OTHER ACFT TO LAND. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE CAPT REITERATED HIS VIEW THAT THE GND CTLR WAS NOT AWARE OF THE ACFT ON FINAL OR HAD LOST TRACK OF HIS POS WHEN HE GAVE CLRNC TO CROSS RWY 4R. HE ALSO STATES THAT THE PACE OF OPS AT MIDWAY ARPT CAN BE DISCONCERTING TO AN OUTSIDER.