  For a smooth, projective, complex algebraic variety $X$, the Riemann--Hilbert
correspondence establishes a complex analytic isomorphism between the `Betti
moduli space' of rank $n$ local systems on $X^\mathrm{an}$ and the `de Rham
moduli space' of rank $n$ vector bundles with flat connection on $X$. In the
rank one case, C. Simpson precisely characterizes the subvarieties of these
moduli spaces that are `bi-algebraic' for this typically transcendental,
analytic isomorphism. In this short note, we give a new proof of this
characterization of Simpson, using methods from o-minimal geometry. We adapt
the o-minimal proof to a p-adic setting, namely that of Mumford curves.
