Let $G$ a real algebraic group and $T$ a borel subgroup, $G_C$ be the complexification of $G$ and $B$ be the borel subgroup of $G_C$ corresponding to $T$. In Woit's Borel Weil Bott's notes, Woit claims that $G_C/B = G/T$(he cryptically says as spaces of right cosets.) I don't think this is true even on the level of sets.
Let $G=GL(2, \mathbb{R})$. Then $G/T\cong P(\wedge^1(\mathbb{R}^2))$ where this is an isomorphism of quasi projective varieties, and where $P$ denotes projectivization. $G/T$ is a priori given the structure of a quasi projective variety by taking an action on a well chosen projective space and using the fact that the orbits are locally closed. Similarly $G_C/B\cong P(\wedge^1(\mathbb{C}^2))$.
On the level of sets we have explicitly if $e_1, e_2$ are canonical bases for $\mathbb{C}^2$, that $G(e_1 ,e_1 \wedge e_2)$ is the full flag variety on $\mathbb{R}^2$ sitting in the flag variety of $\mathbb{C}^2$. The Borel, $T$ of $G$ is the stabilizer of this point and is the real upper triangular matrices. If we take $G_C(e_1 ,e_1 \wedge e_2)$ this is the full flag variety on $\mathbb{C}^2$ and the stabilizer of this point is exactly the $B$ that corresponds to $T$. The orbit under $G_C$ strictly contains the orbit under $G$.