Article 22 of Immigration Act says
“An alien who possesses a valid passport with a valid visa, a valid passport without a visa, or a valid travel document, shall acquire the right to stay or reside after having been inspected by National Immigration Agency, and allowed to enter the State.
An alien who acquires the permission to stay in the State pursuant to the preceding Paragraph shall apply to National Immigration Agency for an Alien Resident Certificate within fifteen (15) days after the date of the entry.
I think it could be read that a foreigner who entered in Taiwan using a visa has been permitted to stay or reside already, and who has gotten a permission to reside should apply for an ARC within 15 days.
My comprehension of both of English and Chinese is questionable, though.