It might sound odd, but when you think about it you'll find many clubs who play outside of their country. Canadian clubs in the MLS, AS Monaco in France, Wrexham AFC in England are all well known examples, but there's one club that goes a step further.
In 2006 Australia moved from the Oceania Football Confederation to the Asian Football Confederation due to the enormous disparity in quality it had with the other OFC members. A 31-0 win over American Samoa made this abundantly obvious. Just one year later, Wellington Phoenix, New Zealand's only fully professional club, made it's debut in the A-League.
This makes Wellington Phoenix the only club in the world that not only plays in a different country, but also in a confederation that it is not a member of. This was not without controversy, the AFC president stated that Wellington Phoenix had until 2011 to permanently move to Australia due to AFC criteria. FIFA president Joseph Blatter stepped in and said that if the club and the league were in agreement then they had FIFA's blessing and that was enough, the confederation could not intervene.
With that settled, there is still to this day one "issue" at hand, Wellington finds itself in a limbo in terms of continental competitions. Despite their participation in the A-League, they are a guest in the Asian Confederation and therefore cannot qualify to any Asian continental competition. Also, because the only way New Zealand allocates OFC Champions League spots is through the local league that Wellington Phoenix does not participate in, they cannot play in Oceanian continental competitions either.
I wrote "issue" above within quotations because the club is and always has been aware of the situation. At the end of the day, it's a concession they're willing to make in order to play in a more competitive and fully professional league.
Who knows, maybe one day they'll actually play in New Zealand, maybe they'll get accepted by the AFC, or maybe New Zealand as a whole will move to Asia like Australia did in 2006. Until then Wellington Phoenix's continental ambitions will have to wait.