I seem to have lost the reference to a realisation I am interested in. Hopefully someone can steer me to a paper that fully explains the realisation.
For the case $K_2$(the 5-gon) the following process is taken to give the realisation.(Although one crucial step is missing in my memory!)
Consider the triangulations of the $5$-gon and consider now a particular triangulation. In each of the 3 regions cut out in this triangulation put a point. We label the points $1,2,3$. We do this for each of the triangulations, however the information forgotten is exactly how we allocate these numbers (to the triangulated regions) for all the triangulations.
Now we may orient all the triangulations so that we can speak about a common boundary arc. Define a co-ordinate associated to the triangulation as follows:
through the "common" chosen boundary arc draw a line to the vertex inside the first triangle it meets, then draw two more such lines and so on. (These lines will either go to another vertex inside an adjacent triangulation or through a boundary arc.)
Denote the co-ordinate associated to a triangulation by $(x_1,x_2,x_3)$ where x_1 is obtained by considering the vertex $1$. If there are $m$ paths passing through boundary arcs when you take the line left of $1$, and $n$ paths passing through boundary arcs when you take the line right of $1$ then set $x_1 = nm$.
For the 5-gon we get the co-ordinates (4,1,1), (3,2,1), (3,1,2), (2,1,3), (1,2,3).
These points lie on a plane in $\mathbb{R}^3$, so we do indeed get $K_2$.
I was hoping someone knows where I can find a paper describing the realisation of K_n by considering triangulations of the n+2-gon in the way outlined above.