I have used far too many RTWs in parallel. I have made best efforts to make sure that the rules of each ticket are not violated (and none of the rules stated that I cannot have other air travel while a current itinerary is in progress). I have had them on overlappinng OWs (issued by different airline partners and on different PNRs) and Star and other partners. Never had a problem and I specifically asked RTW desk once and they said no probs.