Is it known whether any **projective geometry statement** that holds true in the real projective plane (equivalently, can be deduced from Hilbert axioms) follows from the **standard projective axiomatics**?


By **projective geometry statement** I mean any statement it terms of incidence of points and lines, that is a statement of the form: 
Let $I\subset \mathbb{N}\times\mathbb{N}$ be a finite subset and $(i\_0,j\_0)\in\mathbb{N}\times\mathbb{N}$.
For all sequences $(p\_i)$ of points and sequences $(l\_j)$ of lines, 
if $p\_i\in l\_j$, for all $(i,j)\in I$ then $p\_{i\_0}\in l\_{j\_0}$.


By **standard projective axiomatics** I mean four incidence axioms
plus Desargues, Pappus and Fano.



Does any real projective plane incidence theorem follow from axioms?
This is false. I don't see how to get a counterexample from Andreas Blass's comment (the only uses for the existence of $\sqrt2$ which are obvious to me require a more flexible notion of incidence statement), so I am posting this as an answer although it is probably more complicated than necessary. 


For fixed prime $p$, one can take a point and line configuration corresponding to the rank 3 Dowling geometry of $\mathbb{Z}/p$, in which a certain equality must occur unless the underlying field has a primitive p-th root of unity. The presence of a root of unity of prime order $p \ge 3$ is not ruled out by these axioms, which can therefore never prove the equality, although it holds over $\mathbb{R}$.


Explicitly, what you do is the following: take 3 points and the 3 lines connecting them (think of this as a triangle). On each of these three lines, place $p$ points, labelled $0,1,\ldots,p-1$. Label the lines by $a,b,c$, and denote by $i\_a, i\_b, i\_c$ the point labelled $i$ on the corresponding line. Define a new line on which $i\_a, j\_b, k\_c$ are present whenever $i+j+k = 0 \mod p$.


Any realization of this point and line arrangement over a field, in which the corners of the triangle are all distinct, corresponds to a representation of $\mathbb{Z}/p$ in the multiplicative group of the field. Hence if $p\ge 3$, either some two corners coincide or the representation is trivial, so all points $\{i\_a\}\_i$ are identical, as are all points $\{i\_b\}\_i$ and all points $\{i\_c\}\_i$. 


The many lines connecting various triples of points also imply that if two corners of the triangle coincide, so do all points $i\_a, j\_b, k\_c$ for all $i,j,k$. 


Hence in $\mathbb{R}$, one has that $1\_a$ lies on the line through $0\_a, 0\_b, 0\_c$. This incidence statement is not true over $\mathbb{C}$, hence it is not implied by the axioms given.


