In my research, one can find as a special case the following family of varieties. Fix integers $0<k<n$ and let $G=Gr(k,n)$ be the Grassmannian of $k$-planes in an $n$-dimensional vector space $V$ (the base field doesn't really matter, but for me it is $\mathbb C$). Fix a basis for $V$ and consider the matrices $\theta_i=\mathrm{diag}(1,\dots,1,0,\dots,0)$ of rank $i$ (equivalently, we fix a flag and consider the projection maps onto the flag). Then let $\Phi(k,n)$ be the set of $(P_1,\dots,P_n)\in G^n$ such that $\theta_1\cdot P_1 \subseteq \theta_2\cdot P_2 \subseteq \cdots \subseteq \theta_n\cdot P_n =P_n$.
For $k=1$ and $n=2$, this gives a wedge sum of two projective lines.
Clearly this is related to flag varieties. It seems like it might be a Springer fiber; for instance, the fiber of the matrix $ \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} $ in $SL_3$ is also a wedge sum of two projective lines. But I wonder if this is also a more antiquated family of varieties known to combinatorial geometers?
