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Get a police report saying you lost your APRC.
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TECO might not be able to issue a new APRC, only NIA can do that, but they can authenticate the above mentioned report which can be handy not only to get a new card here in Taiwan but also on the plane.
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Call NIA in Taiwan - Skype is handy, or any other cheaper alternative, not landline- and inform them of your situation.
https://www.immigration.gov.tw/5475/5478/141465/141808/141970/
Previous cases:
https://www.reddit.com/r/taiwan/comments/18a2dcj/lost_arc_in_foreign_country/?rdt=41043