Kansas City ARTCC (ZKC) had cleared Ozark Flight 709 (a DC-9, N923L) to cruise at FL270. While proceeding ESE near the edge of the ZKC area, the DC-9 converged on Zorro 11 (a flight of 3 F-4's). The F-4's had been cleared by Indianapolis ARTCC (ZID) to proceed westbound at FL260 to rendezvous with a KC-135 tanker (Shake 81) to refuel on aerial refueling route AR110W. The beginning of AR110W was in the ZID area; from there it continued westbound into the ZKC area. The tanker was orbiting under control of ZKC and was cleared to maintain block altitude FL260 to FL290. ZID cleared the F-4's to the tanker frequency without handing them off to ZKC and without obtaining approval from either ZKC or the tanker. After establishing radio contact, the tanker crew incorrectly cleared the F-4's to climb to FL270. When the ZKC controller realized the F-4's and DC-9 were converging, he told the DC-9 crew to immediately turn left. The DC-9 crew saw the F-4's and took evasive action. ZID had not properly coordinated the F-4s' position and none of the pilots were informed of conflicting traffic.