I had a similar thing happen to me flying TYS-MSP-NRT-ICN-CJU. The NRT-CJU legs were Korean air, and the TYS-NRT was Delta. Delta wouldn't show the ICN-CJU leg on there website. Korean only showed the MSP-CJU legs on their website. It still worked, though I will say it was messy in NRT both ways. Korean air didn't like the way Delta had tagged my bags and vice-versa, so each time it took about 30 minutes to get that fixed. As far as checking in, I checked in with Delta and they gave me the regular boarding passes for their flights. They also gave me two pieces of paper for the Korean flights that said to exchange for boarding passes at the gate.