The DJ construction works with a simplicial complex $K$ and a subtorus $W\leq\prod_{v\in V}S^1$ (where $V$ is the set of vertices of $K$). People tend to be interested in the case where $|K|$ is homeomorphic to a sphere, but that isn't really central to the theory. However, it is important that we have a simplicial complex rather than something with more general polyhedral structure. It is also important that we have a subtorus, which gives a sublattice $\pi_1(W)\leq\prod_{v\in V}\mathbb{Z}$, which is integral/rational information. I don't think that the DJ approach will help you get away from the rational case.
I like to formulate the construction this way. Suppose we have a set $X$ and a subset $Y$. Given a point $x\in\prod_{v\in V}X$, we put $\text{supp}(x)=\{v:x_v\not\in Y\}$ and $K.(X,Y)=\{x:\text{supp}(x) \text{ is a simplex}\}$. The space $K.(D^2,S^1)$ is a kind of moment-angle complex, and $K.(D^2,S^1)/W$ is the space considered by Davis and Januskiewicz; it has an action of the torus $T=\left(\prod_{v\in V}S^1\right)/W$. Generally we assume that $W$ acts freely on $K.(D^2,S^1)$. There is a fairly obvious complexification map $K.(D^2,S^1)/W\to K.(\mathbb{C},\mathbb{C}^\times)/W_{\mathbb{C}}$. Under certain conditions relating the position of $W$ to the simplices of $K$, one can check that $K$ gives rise to a fan, that the complexification map is a homeomorphism, and that both $K.(D^2,S^1)/W$ and $K.(\mathbb{C},\mathbb{C}^\times)/W_{\mathbb{C}}$ can be identified with the toric variety associated to that fan.
This falls slightly short of a proof but is too long for a comment.
The question inquires into the largest family of triangulations in which every pair of members shares some diagonal.
If we instead fix any single diagonal in common to every member of a family of triangulation of an n-gon, it is self-evident (from the noncrossing property of triangulations) that the number of triangulations sharing it is given by the product of the enumerations of the two families of triangulations of the ploygons so formed either side of it.
For any triangulatable n-gon, any given diagonal is a member of up to two fan triangulations, in which it can be assigned an integer the $d^{th}$ diagonal from the perimeter satisfying $1\leq d\leq \frac{n-2}{2}$. By symmetry we may choose either of the two fans with no effect upon the distance of the diagonal from the nearest edge.
$d=\frac{n-2}{2}$ shall be the distance of the central triangulation if $n$ is even and $d=\frac{n-3}{2}$ shall be the distance of the two "most central" diagonalisations if $n$ is odd.
The number of triangulations of any $n$-gon sharing the $d^{th}$ diagonal is the product of the number of triangulations $\cal T(d,n)$ either side of it:
$$\cal T(d,n)=C_d\times C_{n-d}$$
$$\cal T(d,n)=(d+1)^{-1}\binom{2d}{d}(n-d+1)^{-1}\binom{2n-2d}{n-d}$$
The ratio of one Catalan number to the next is greater than the ratio of the previous two Catalan numbers, and therefore in the choice of which diagonal to share, in order to maximise the size of the family, for any given choice $d$ of diagonal which is not at the edge, we may always identify a larger family by choosing the diagonal $d-1$ which is one position closer to the perimeter and by induction the outermost diagonal $d=1$ yields the largest family.
The size of the largest family of triangulations sharing the same diagonal is therefore given by:
$|\cal S| = C_{n-2-1}\times C_1=|\cal T_{n-1}|$
My claim which would complete the proof (and which remains to be proven) is that there is no family of triangulations in which every pair shares some diagonal, which is larger than the largest family of triangulations in which every triangulation in the family shares the same diagonal.
I suspect this is proven by the fact that the largest face on any associahedron has the same number of edges as the number of vertices of the associahedron one order smaller. In particular I believe that each face of the associahedron represents a family of triangulations which all share the same edge. These two statements alone would in fact be sufficient to answer the entire question in the affirmative.
Best Answer
There is quite a lot that can be said about the Charney-Davis conjecture, so let me say a few things and perhaps add more later.
1) The conjecture is a discrete analogue of a well known conjecture by Hopf on the Euler characteristic of nonpositively curved manifolds in odd dimension. You consider cubical complexes and the condition of nonpositive curvature is replaced (the CAT version) by the condition that all links of vertices (which are simplicial spheres to start with) are flag. (Namely they are the clique complexes of their 1-skeletons.)
2) The original formulation is in terms of the h-vectors and it says that if K is a flag complex which is a sphere of dimension d-1, and d=2e, then
I suppose that this is equivalent to Matt's formulation in terms of the face number $f_i$. ($f_i(K)$ is the number of i-dimensional faces of K.)
3) The conjecture as is, should extend to the case where K is a Gorenstein* complex (i.e. a homology sphere in the weakest possible sense with coefficients in a field k).
4) The conjecture is not known to hold for th case where K is the boundary complex of a simplicial polytope!
5) Stanley's book on combinatoris and commutative algebra (second edition p.100) is a good source (but a lot have happened later).
Let me mention two remarkable case where the conjecture is proven.
6) For 3-dimensional flag spheres. This is an extremely difficult result by Davis and Okun.: Michael Davis and Boris Okun, Vanishing theorems and conjectures for the $\ell^2$-homology of right-angled Coxeter groups. Geom. Topol. 5 (2001), 7–74. Archive version link text. The proof relies on a 1995 deep results by J Lott and W L¨uck, $L^2$ –topological invariants of 3–manifolds, Invent. Math. 120 (1995) 15–60.
For 3-dimensional flag triangulations of spheres the Charney-Davis conjecture asserts that $f_1 \ge 5f_0 -15$. Barnett's lower bound conjecture asserts that for every triangulation of 3-dimensional sphere $f_1 \ge 4f_0-10$ and it is known that this statement extends to triangulated manifolds and pseudomanifolds. Likewise perhaps the 3-dimensional Charney David conjecture, namely Davis-Okun's theorem, extends to arbitrary flag-triangulated pseudomanifolds.
7) For certain polytopes described by a geometric condition on their fans which is stronger than being flag. This is a remarkable result by N.C. Leung and V. Reiner, "The signature of a toric variety", Duke J. Math., 111(2002), 253-286. Preprint (ps file).