Well, as I noted in my TL (obligatory shameless plug) there was an important faction in Korean political thought in the 17th and 18th centuries that was inspired by Italian Jesuits. That's the only link I can think of offhand.
Or boringly, Korea stays a vassal of China (Japan doesn't modernise, etc.) and is included as part of the Chinese coastlands in some colonial carve-up that includes united Italy as one of the European colonial powers.