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According to the InterWebs there are only two Korean restaurants in Romania, both in Bucharest (Dami and Korea-house), however where-ever there is a good number of Korean tourists or workers, there is usually also a Korean restaurant.
Looking at the web site of Daewoo Mangalia Heavy Industries. all the managers listed there have Romanian names, the president and CEO is Korean, but it seems he is based in Korea. So we can assume there are not many Koreans working at the site, and therefore there is no place to get Korean food.
There are a few Chinese places up the coast in Constanta.