That made me realise how often characters are (randomly) loaded! eh eh
For me:
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Another Miss Oh : Both characters are middle class, she's an office worker, he owns a little sound effect company that he needs to keep afloat, they're both renting. (and it's a good show)
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Ms Temper & Nam Jung Gi: main characters are middle class or upper middle class, it's very realistic when it comes to character's wealth too. And just a very very good drama.
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Secret Love Affair: The male lead is on the poor side while the female lead (and her husband) are more well off, but it doesn't really get overboard with it. Rich people are all around though, but it serves the plot well, it's not just for show.
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Witch's Romance: The female lead is a journalist, the male lead is doing odd jobs. There's a bit of money appearing at the end, but it's not crazy either way.
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Bring It On, Ghost: no rich main character (that "company" car they end up with is proof enough)
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High School King of Savvy : main characters aren't from rich family, kind of the opposite
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Second to Last Love : main characters are upper middle class or so
I'm not 100% sure those still fit the bill:
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City Hall: he obviously has some money, but it doesn't seem too crazy
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Drinking Solo: The male lead has money because of his job (but let's be honest, it looks like too much money for such a job, no matter how much of a "star" he is).
To rebound on Panda's suggestions:
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It's okay that's love: she may not be that well off, but he owns more than one (huge) apartment/house, it's not a direct plot point, but it's still clear through the show that he has much more money.
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radiant office: yes! The male lead has a good job and therefore some good money (enough that he is not afraid to leave a job if needed), but it's still realistic compared to his status and not played upon.
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laurel tree tailors/The Gentlemen of Wolgyesu Tailor Shop : Main male lead has money due to his past job (it shows mostly at the start), but it's not unrealistic, a few other characters are from a CEO family, but compared to the bunch of character, it's a good little ratio. The lack of money is actually part of the plot several times and it's always well used.
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coffee prince: yes it's a classic, but his family has lots of money... so much that they place him at the head of a café, while they give him limited income to make it work, it's still obvious he's loaded...
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let's eat 1 and
2: yes! I won't speak for the second one for sure, but first is just full of middle-class characters.