  The character variety, which parametrizes reductive representations of
finitely generated groups in $\mathrm{PSL}_2(\mathbb{R})$, has many
compactifications. We construct a continuous surjection from the real spectrum
compactification to the oriented Gromov--Hausdorff compactification of the
character variety. For this purpose, we assign an orientation to the real trees
appearing in the boundary of the real spectrum compactification using that the
usual orientation on the circle is described by a semi-algebraic equation.
Moreover, we show that the orientation on the real trees which appear in the
boundary of both compactifications are well understood by the study of
sequences within the character variety that converge toward the points in the
boundary.
