Thanks for oleo436, Matt.Vargo, X X X, PIPPY and RayRaces among others for their patience and help to develop my theory.

Observations:

1. From Firemonkey's Zendesk Request Form, there are two choices for Region.
   1. North America, Canada or Japan
   2. Europe, Africa, Asia Pacific or South America
2. Going to the address of Real Racing 3 in the US iTunes App Store then changing the Store to Hong Kong does not redirect to the corresponding page of Real Racing 3 in Hong Kong
3. Users of games downloaded from an American iTunes Store can connect with each other without any problem but those who downloaded elsewhere cannot
4. Going to the Game Centre and clicking into a friend's name revails the friend's games. In that page, if the balloon of "Points" is clicked, it brings to a page to compare the points from various games. For users in the same region, Real Racing  is recognized as a single game and compared. For users from different regions, this comparison shows two rows, signifying that Game Centre recognizes Real Racing 3 as two different games just because the game is obtained from different regions.

Therefore, I would hypothesize that the packaging of the game when we download the game contains certificates or information regarding the entry in the Game Centre as well as the iTunes App Store. For some reasons, the information from the two regions are different, making the Game Centre recognize it as two different games. That is why players from the first region cannot race with the second, vice versa.

Experiments:

I would need some volunteers to test it out. There will be three components:

1. Player's physical location
2. The App Store the game is downloaded from
3. The Apple ID used in Game Centre

The tests should contain:

1. All the same
2. Physical location in different regions, same region for Store and Game Centre
3. Physical location in same region, different regions for Store but same for Game Centre
4. Physical location in different regions, different regions for Store but same for Game Centre
5. Physical location in same region, same region for Store but different for Game Centre
6. Physical location in different regions, same region for Store but different for Game Centre
7. Physical location in same region, different regions for Store and Game Centre
8. Physical location in different regions, different regions for Store and Game Centre

We probably can't test out all the cases, so basically, please post if you have any experience on connecting with your friends, may it be successful or not. We will try to deduce if the theory is indeed accurate.