Let $\Sigma$ be a Riemann surface and let $n,d$ be two relatively prime integers. We can consider different moduli spaces related to those. On one hand we have:

-$M_{Dol}$ the moduli space of stable Higgs field of rank $n$ and degree $d$

-$M_B$ moduli space of (twisted) representations of fundamental group of $\Sigma$ with central monodromy $e^{\frac{2 \pi d}{n}}$

-$M_{dR}$ moduli space of stable meromorphic connections with one pole with residue $-\frac{2 \pi i d}{n}$.

In the case $n=1$ there is an explicit description of the first twos:

$$M_{Dol}=Jac(\Sigma) \times \mathbb{C}^g $$ and $$M_B=(\mathbb{C}^*)^g .$$

I struggled to find an explicit description for de Rham moduli space however. I know this is biholomorphic to $M_B$ but it's not the same as an algebraic variety. Is there a well known algebrogeometric description of it? Is there a concrete description of the Riemann Hilbert map too in this case?

EDIT: The general question was motivated by the following way more specific one. What happens at the moduli space if we consider a map (not necessarily holomorphic) $f:\Sigma \to \Sigma$ and we look at the map on the moduli space $$(\mathcal{E},\nabla) \to (f^*\mathcal{E},f^*\nabla) .$$

I was trying to get a glimpse of the geometrical properties of this kind of morphism(for example I think it should be holomorphic but I can't see why) via the identifications in the answer(which are very well written anyway).