I attended a talk which generalized matroid realizability over a field to matroid realizability over division rings, and showed that the question of realizability is undecidable. However, they used a word problem arising from the division ring. 

Is it known whether the question of "Is a matroid M realizable over any field F" is computable? 

It seems to me that it shouldn't be, because they you would have a prover of all theorems akin to the Pappus theorem, but that's just my intuition.