How much does the existence of hyperplane arrangements depend on the underlying field?

Question

Given an arrangement of hyperplanes $H_i$ in $V$ a vector space over a field $k$, we have the associated incidence structure. Not all realisable incidence structures can occur over all fields however, a stupid example is taking more hyperplanes $H_i$ than the total number of hyperplanes when over a finite field. I am wondering if there are more subtle realisability obstructions, eg, is $k$ infinite sufficient to realise all realisable incidence structures? What about $k$ algebraically closed? This seems like a very natural question, so I'm sure someone has thought about it before, so a reference would be much appreciated.