When can splits be drawn in the plane?

Split networks are a popular tool for the analysis and visualization of complex evolutionary histories. Every collection of splits (bipartitions) of a finite set can be represented by a split network. Here we characterize which collection of splits can be represented using a planar split network. Our main theorem links these collections of splits with oriented matroids and arrangements of lines separating points in the plane. As a consequence of our main theorem, we establish a particularly simple characterization of maximal collections of these splits.