Hi Mi,
If you are a Korean native (gyopo) then you may qualify for the F-4 visa. Yes, you are eligible.

Please submit your online application at www.parkenglish.com.

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F-4 Visas Process for people with Korean Ancestry

Those with Korean ancestry (“Gyopo"), must apply for an F-4 Visa after they arrive in Korea, not before.

The F-4 application requires the mother or father’s “Ho Juk Deung Bon" (a copy of this form is
available at the “gu" office in South Korea where the parents lived prior to leaving Korea)

To apply for the form the following information is needed:

-Mother/Father’s full name
-Mother/Father’s old Korean ID #
-Mother/Father’s old Korean address

Once the form has been obtained it must be taken to the Korean Immigration Office, along with the following documents:
-A copy of your naturalization papers or foreign birth certificate.
-Passport
-4 recent passport size color photographs.
-Current resume

There is a visa processing fee (check the Korean Immigration Office for details *below), and it takes about a week to process. The F-4 visa allows you to live and work in Korea much like a native. The visa is renewable every two years, and you don’t have to change your visa if you change jobs.

*Immigration Site: http://www.immigration.go.kr/indeximmeng.html