  In this article, I define triangulated categories of constructible
isocrystals on varieties over a perfect field of positive characteristic, in
which Le Stum's abelian category of constructible isocrystals sits as the heart
of a natural t-structure. I then prove a Riemann-Hilbert correspondence,
showing that, for objects admitting some (unspecified) Frobenius, this
triangulated category is equivalent to the triangulated category of
overholonomic $\mathscr{D}^\dagger$-modules in the sense of Caro. I also show
that the cohomological functors $f^!$, $f_+$ and $\otimes$ defined for
$\mathscr{D}^\dagger$-modules have natural interpretations on the constructible
side of this correspondence. Finally, I use this to prove that, for any variety
$X$ admitting an immersion into a smooth and proper formal scheme, rigid
cohomology (with lisse coefficients) agrees with cohomology defined using
arithmetic $\mathscr{D}$-modules.
