Upvote:7
During the Second Taiwan Strait Crisis in 1958, "32 Nationalist Chinese F-86s clashed with 100 Red Chinese MiGs in a series of aerial engagements".
American naval aircraft also helped the Nationalist air force establish control of the region's airspace. Nationalist pilots flying American-made fighters defeated their Communist opponents in a series of air battles that cast doubt on the quality of Communist's pilots and aircraft.
http://www.globalsecurity.org/military/ops/quemoy_matsu-2.htm
This was also the conflict that saw the debut of the venerable AIM-9 Sidewinder AAM.
This is probably the only example of a "serious air clash" between PRC and Taiwan. In the closing stages of the Chinese Civil War, the PLA's failure to take small islands like Kinmen exposed serious shortcomings in amphibious capability. The Korean War forced the US to place Taiwan under its protection, and after the cease fire the US turned its attention to Taiwan, sponsoring an invasion of China sparking the First Taiwan Strait Crisis. In reality both sides were woefully incapable of taking over the other - the PRC hampered by its lack of capability and Taiwan's US protection, and Taiwan by its overwhelming numerical disadvantage. The conflict soon reached a stalemate, followed by a gradual thaw in relations over the late 20th century.