  Let $G=\mathrm{GL}_{2n}(\mathbb{R})$ or $G=\mathrm{GL}_n(\mathbb{H})$ and
$H=\mathrm{GL}_n(\mathbb{C})$ regarded as a subgroup of $G$. Here, $\mathbb{H}$
is the quaternion division algebra over $\mathbb{R}$. For a character $\chi$ on
$\mathbb{C}^\times$, we say that an irreducible smooth admissible moderate
growth representation $\pi$ of $G$ is $\chi_H$-distinguished if
$\mathrm{Hom}_H(\pi, \chi\circ\det_H)\neq0$. We compute the root number of a
$\chi_H$-distinguished representation $\pi$ twisted by the representation
induced from $\chi$. This proves an Archimedean analogue of the conjecture by
Prasad and Takloo-Bighash (J. Reine Angew. Math., 2011). The proof is based on
the analysis of the contribution of $H$-orbits in a flag manifold of $G$ to the
Schwartz homology of principal series representations. A large part of the
argument is developed for general real reductive groups of inner type. In
particular, we prove that the Schwartz homology $H_\ast(H, \pi\otimes\chi)$ is
finite-dimensional and hence it is Hausdorff for a reductive symmetric pair
$(G, H)$ and a finite-dimensional representation $\chi$ of $H$.
