The awkward expression there exist.

Heck, you caught me! I tried to avoid this question in my first response.

No, I do not find the phrase "There exist" awkward when used for effect in prose writing, but I do not recommend its use in a formal paper unless "there" is used as an adverb and refers to an actual location:'There exist in the quantum world ....'

Otherwise, the usage tends to drive some professors completely nuts. When "there" is not used as an adverb, we're talking about an "existential construction"—grammatically a very tricky structure. For instance, the verb "hay" in Spanish has no subject, and that's confusing enough. But when the verb "there" is used in the same way in English, it is considered to be—maybe you better sit down first before I tell you this—a Pronoun! No, I'm not kidding!

'There are effects.'
Pronoun + linking verb + noun complement

That said, if you fully understand the construction and can justify its use, then, of course, you're free to employ it. In fact, we sometimes can't avoid it. So, use it sparingly.