In unpublished work, Geelen proved that a matroid is near-regular if and only if it has no minor isomorphic to: U2,5; U3,5; the Fano plane and its dual; the non-Fano and its dual; the single-element deletion of AG(2,3), its dual, and the matroid obtained from it with a Delta-Y operation; and P8. We provide a proof of this characterization.
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Matroid Structure for Efficiency
Networks and Optimization

Hall, R, Mayhew, D, & van Zwam, S.H.M. (2009). The excluded minors for near-regular matroids. arXiv.org e-Print archive. Cornell University Library .