I think when it comes to unorthodox drama, Japan offers more variation. Some of my favorites are:
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Tiger and Dragon, which is about a group of rakugo artists. Rakugo is a Japanese form of one-man comic storytelling.
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Dakara Koya, about an under-appreciated housewife who went on a roadtrip
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River's Edge Okawabata Tanteisha, about a detective agency that caters to clients with weird requests (e.g. finding a one-hit wonder idol from a client's youth)
As for Korean dramas, I recommend:
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The End of the World: viral epidemic meets zombie apocalypse
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Heard It through the Grapevine: I haven't watched this myself, but it's about a teenage couple who decided to keep their baby and how their parents (especially the guy's, who are wealthy and very reputation-conscious) deal with it