My GF lost her Korean passport and she needs to get her family registry, etc.

You said her previous marriage to a US citizen was abusive. VAWA might be a possibility; though that depends on whether there is evidence of actual abuse. (Of course, if you are a US citizen, then applying for a green card based on marriage to you is simpler, but the problem with that is the lack of ID for marriage.)
She’d still need ID for a VAWA application, wouldn’t she?
 
No other ID. Since the Korean government can't really help me I was considering the Asylum angle for her since if she was to go back to Korea she would basically be homeless with nobody to help her there. The Korean Consulate was going to renew her passport but her fingerprints did not match what they had on file which is why they wanted the family registry. The Korean Consulate does believe that she is who she says she is since she knew her Korean identity number (like our SS number) and she could give them her parents and brothers, sisters names etc. She originally came here on a student visa and eventually married a Korean US citizen but the marriage became abusive so she got a divorce. She was trying to go back home but her step mom basically said she is on her own so she has stayed here.
Great
 
She should contact the Korean consulate to explore alternative ways to prove her identity, like using her Korean national ID or other records. For marriage, check with the consulate if other forms of ID are acceptable. If she’s at risk in Korea, she may want to consult an immigration attorney about claiming asylum.
 
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