at some time ago, PRC(China) even did not allow national carriers serve PRC and ROC(Taiwan) at the same time, so there are a lot of "Asia" airways/airlines when those carriers want to serve Taiwan, like British Asia, Qantas Asia , KLM Asia and Japan Asia Airways. Some of them are still running, usually using the call sign and planes of their parents company directly. Also China Airline (CI/CAL, ROC/Taiwan's flag carrier) could not go to Europe as their old livery carry ROC flag. This lead to the set up of Mandarin Airlines(AE/MDA), whose a "private" airline and avoid any political problems when serving Europe. The flags are still important issues, CI changed their livery before 1997 so that no political problem would be created by the ROC flag livery after the hand over of HKG from UK to PRC. Also Air China (CA/CCA, PRC/China's flag carrier) removed PRC flag on some planes or just send those have special liveries to Taiwan flights in order to avoid problems after direct flight begin.
I've to say this benefit HKG a lot, although not good for people at both China and Taiwan. Even now CI own 10% market shares on pax. number at HKG, and giving the golden chance to let CX+KA become one of the leading airline groups in East Asia.