The usual candidates for context free languages whose complement is also context free, but they are not regular are the Deterministic Context Free Languages ($DCFL$).

For example, $L=\{a^nb^n\mid n\in\mathbb N\}$ is such language.

Is there a language, $L\in CFL\setminus DCFL$ such that $\bar L\in CFL$?

Since $DCFL$ is closed under complement, this is equivalent of asking whether $$CFL\cap co\text{-}CFL = DCFL$$


See here:

For every finite, non-unary alphabet, the language of all palindromes is not in DCFL, but in the intersection of coCFL and CFL.


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