If the ROK government provides legal status to the babies of illegal immigrants they will create further incentive for illegal immigrants to come to South Korea. They only have to look at what happened in the U.S. to see what will happen in the ROK:

Multiple bills aimed at establishing a birth registration system for undocumented migrant children born in South Korea remain pending in the National Assembly, leaving thousands of children without legal recognition and access to basic services, despite growing calls for legislative action.
At least two proposals pending in the parliament seek to create a legal framework allowing the registration, certification and amendment of birth records for non-Korean children, who are currently excluded from the country’s family registry system. No law has yet been enacted.
Under existing rules, children born to foreign parents cannot be formally registered at birth even if they are born in South Korea, as the family registry system applies only to Korean nationals. Without a resident registration number, many face difficulties accessing health care, education, child care and social protection.
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