Consider the real diagonal $4\times 4$ - matrix  

$$I\_{2,2}={\rm diag}(1,1,-1,-1)$$ 
and the corresponding special unitary group
$$ G={\rm SU}(2,2)=\{g\in {\rm SL}(4,{\mathbb{C}})\ |\ g\cdot I\_{2,2}\cdot \bar g ^{\rm tr}=I\_{2,2}\}.$$
We regard $G$ as an algebraic group over ${\mathbb{R}}$.
It is known that $G$ is quasi-split, that is, $G$ contains a Borel subgroup $B$ defined over ${\mathbb{R}}$.
Let $U$ denote the unipotent radical of $B$, it is defined over ${\mathbb{R}}$. 
Set $X=G/U$.



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> **Question.** *How can one describe the homogeneous space $X$* ***explicitly*** *(by equations and "inequalities") as a quasi-projective variety over ${\mathbb{R}}$ on which ${\rm SU}(2,2)$ naturally acts?*
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EDIT: I explain why I need explicit equations for $G/U$. I want to *twist* the desired variety by the 1-cocycle $c=I\_{2,2}\in G(\mathbb{R})\_2$ and to obtain explicit equations for the twisted variety $Y={}\_c X$. 
This new variety $Y$ over $\mathbb{R}$ is a homogeneous space of the twisted group $\_cG={\rm SU}(4)$ for which $U$ is the stabilizer of a $\mathbb C$-point. 
Of course, $Y$ has no $\mathbb R$-points, because the stabilizer of an $\mathbb R$-point would be a compact form of $U$, but we know that the unipotent group $U$ has no compact forms.



Explicit description of SU(2,2)/U
I guess your variety is just the variety of pairs of ${\mathbb C}$-linearly independent vectors in ${\mathbb C}^4$ that are isotropic with respect to this Hermitian form and orthogonal to each other. Respectively, the twisted form is the same variety but for another Hermitian form (and if the form is not hyperbolic, there is no such a pair over ${\mathbb R}$, as you mentioned).


