The best example I can think of is American. They have Amadeus running their international website. Their ticketing system is internal. And all their inventory is managed in Sabre.
Southwest was similar, for a while Amadeus ran their international site while Southwest ran an outdated internal system for domestic travel that didnt support flights leaving and arriving on different days. They eventually had Amadeus move into running their domestic stuff a few years ago and now they have red-eyes.
Delta runs all their own stuff on a mainframe and from the outside it looked like a slow moving disaster. I know Amadeus sees all their inventory and looks at each passenger.