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.