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. It simultaneously generalizes the Riemann-Hilbert correspondence for germs of meromorphic connections of exponential type, and that for mild exponential difference modules. 

As an application, we prove a variant of the comparison of isomorphisms theorem for a simple class of algebraic 1-forms on complex curves.