A Riemann-Hilbert correspondence for cohomology theories of closed 1-forms

Motivated by the work of Kontsevich-Soibelman on the comparison of isomorphisms conjecture for closed algebraic $1$-forms, we establish a Riemann-Hilbert correspondence of Deligne-Malgrange type.
As an application, we prove a variant of the comparison of isomorphisms theorem for a simple class of algebraic $1$-forms on complex curves.