Let $C_g$ be a compact Riemann surface of genus $g$, and let $X:=\mathrm{Sym}^2(C_g)$ be its double symmetric product and $\Delta \subset X$ the diagonal.

Question. What is the topological fundamental group $\pi_1(X - \Delta)?$

I'm primarily interested to the (probably not too difficult) case $g=2$, but I would like to know the general answer. I'm aware that this is probably standard material for the experts, so any reference will be greatly appreciated.