As pointed out by Sam Hopkins in a comment above, secondary polytopes can be seen as a particular case of the fiber polytopes of Billera and Sturmfels (https://doi.org/10.2307/2946575).
As such, every point in the secondary polytope of a point configuration $A=\{a_1,\dots,a_n\}$ corresponds (not uniquely) to a section of the natural projection \begin{align} \Delta^{n-1} &\to \operatorname{conv(A)}\\ e_i &\mapsto a_i, \end{align} where $e_1,\dots,e_n$ are the standard basis in $\mathbb R^n$ and $\Delta^{n-1}$ is their convex hull, that is, the standard $n-1$-simplex.
Each vertex of the secondary polytope corresponds to a unique, extremal, section, obtained by ``coherently'' picking the vertex of each fiber in a fixed direction. This is why regular triangulations are also sometimes called coherent.