I did some research into this. The relevant law is the Employment Services Act article 48 as amended on May 16, 2003. You can find the ESA in Chinese: evta.gov.tw/english/lawevta. … wevta1.htm
Article 48 says that people who do not require a work permit include “one who marries a national of the R.O.C. with domestic residence and a permit to reside.” The Chinese version says “在中華民國境內設有戶籍之國民結婚，且獲准居留者，不須申請許可” which sounds like basically the same thing. Neither one specifies that an ARC is needed, and to my mind both an ARC and an APRC are a “permit to reside”. Of course how this is interpreted in individual cases is the big question. I wouldn’t rule out someone trying to claim that an APRC is different, but the law does not say that.
Article 51 specifies that APRC holders can apply for an open work permit/personal permit.
They did not require this at my last renewal. Besides, the health checkup was pretty basic and not worth it to me for the amount charged. The clinic down the street does a much more thorough checkup for a lot less. The health checkup is mainly looking for serious infectuous illnesses, drug use or (for females) pregnancy.