How do you say 'unity in variety' in Korean?

As Charles Reis Ribeiro noted, "다양성 속의 통일" would be the DIRECT translation. However, according to the Merriam-Webster dictionary, it is a principle that aesthetic value or beauty in art depends on the fusion of various elements into an organic whole which produces a single impression.

"다양성 속의 통일" does not mean this. There is not much meaning to this phrase though; you simply (directly) translate it. It could be similar to "unity in variety," but it's little different. As you might know, since Korea is divided along the 38th parallel, the phrase is used most often in the context of unification of North Korea and South Korea, which is little different from how "unity in variety" is used in everyday life.

On top of my head, I cannot really think of any phrase that matches with "unity in variety."

I think the term could be translated as "다양성 속의 통일", which means "unity in diversity/united in diversity" (by the way those are the official motto of Indonesia and the European Union respectively).