  The Las Vergnas' strong map conjecture, states that any strong map of
oriented matroids $f:\mathcal{M}_1\rightarrow\mathcal{M}_2$ can be factored
into extensions and contractions. The conjecture is known to be false due to a
construction by Richter-Gebert, he find a non-factorizable strong map
$f:\mathcal{M}_1\rightarrow\mathcal{M}_2$, however in his example
$\mathcal{M}_1$ is not realizable. The problem that whether there exists a
non-factorizable strong map between realizable oriented matroids still remains
open. In this paper we provide a counterexample to the strong map conjecture on
realizable oriented matroids, which is a strong map
$f:\mathcal{M}_1\rightarrow\mathcal{M}_2$, $\mathcal{M}_1$ is an alternating
oriented matroid of rank $4$ and $f$ has corank $2$. We prove it is not
factorizable by showing that there is no uniform oriented matroid
$\mathcal{M}^{\prime}$ of rank $3$ such that
$\mathcal{M}_1\rightarrow\mathcal{M}^{\prime}\rightarrow\mathcal{M}_2$.
