A prismatic Riemann-Hilbert correspondence

We study the category of constructible $p$-Kummer-étale sheaves on the log adic generic fiber $(\mathrm{X},{\mathcal{M}}_{\mathrm{D}}{)}_{\eta }$ for the stratification induced by a horizontal normal crossings divisor $\mathrm{D}$, showing that it is antiequivalent to a category of finitely generated unit Laurent crystals on the log prismatic site. We give a local description of finitely generated unit crystals on a framed log affine in terms of $(\phi ,\mathrm{\Gamma })$-modules, and show pushforward compatibility for strict log smooth proper morphisms.