From what I understand, there exists a notion of derived $\mathbb{C}$-analytic space.

Let $T_{an}$ be the pregeometry in the sense of Lurie whose underlying $\infty$-category is the category of open sets in $\mathbb{C}^n$ and whose admissible morphisms are open immersions. Then a derived $\mathbb{C}$-analytic space is a $T_{an}$-structured $\infty$-topos $(X, \mathcal{O}_X)$ satisfying some requirements (see the works of M. Porta).

Moreover, there exists a notion of symplectic structure on derived stack (for example, see the survey of D. Calaque).

My naive question is: has anybody tried to figure out whether it's possible to define compatibility relations between complex and symplectic structures in the derived setting analogous to the classical $\omega(Ju, Jv)=\omega(u, v)$, $\omega(u, Ju)>0$? In other words, has anybody tried to define derived Kähler manifolds?