Let $M$ be a matroid admitting a coordinatization over a complex vector space. If we know that the complex coordinatization space for $M$ is connected, then may we conclude that the matroid admits a coordinatization over the real numbers?


The only examples that I am able to construct which do not have real coordinatizations have disconnected coordinatization spaces. 


Note: Some texts refere to 'coordinatization over a vector space' as 'realizability over a vector space.'



Realization space of matroids
The answer should be no. Here is the reason: It is perfectly possible to have a connected algebraic variety, defined over $\mathbb{R}$, which has no $\mathbb{R}$-points. For example, $\{ (x,y) : x^2+y^2=-1 \}$. Using [Mnev's universality theorem](http://en.wikipedia.org/wiki/Mnev%2527s_universality_theorem), you should be able to build a matroid whose realization space is [stably equivalent](http://en.wikipedia.org/wiki/Mnev%2527s_universality_theorem#Stable_equivalence_of_semialgebraic_sets) to this variety, and thus has the same property. I have not checked the details here.


