  The symmetric strict implication calculus $\mathsf{S^2IC}$, introduced in
[5], is a modal calculus for compact Hausdorff spaces. This is established
through de Vries duality, linking compact Hausdorff spaces with de Vries
algebras-complete Boolean algebras equipped with a special relation. Modal
compact Hausdorff spaces are compact Hausdorff spaces enriched with a
continuous relation. These spaces correspond, via modalized de Vries duality of
[3], to upper continuous modal de Vries algebras.
  In this paper we introduce the modal symmetric strict implication calculus
$\mathsf{MS^2IC}$, which extends $\mathsf{S^2IC}$. We prove that
$\mathsf{MS^2IC}$ is strongly sound and complete with respect to upper
continuous modal de Vries algebras, thereby providing a logical calculus for
modal compact Hausdorff spaces. We also develop a relational semantics for
$\mathsf{MS^2IC}$ that we employ to show admissibility of various $\Pi_2$-rules
in this system.
