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According to Timatic, the tool used by airlines to check document requirements (courtesy of Emirates), you do not need a visa to enter the Republic of Korea and can stay for up to 30 days. Note that, should you not be staying in Taiwan but transiting, you're limited to 3 days in South Korea.
Korea, Republic of (KR)
Visa Exemptions:
Nationals of Philippines with a residence permit issued by USA and a confirmed onward ticket for a flight to USA within 30 days. The following itineraries are possible:
- arrive from a third country and departing to USA (e.g. PVG-ICN-USA); or
- arrive from a third country and depart to another country for a maximum stay of 3 days, before departing to to USA (e.g. PVG-SGN-ICN-USA); or
- arrive after transiting through a third country for a maximum of 3 days (e.g. PVG-ICN-SGN-USA).
Chinese Taipei (TW)
Visa Exemptions:
Nationals of Philippines with a normal passport for a maximum stay of 14 days.