Matroids with an infinite circuit–cocircuit intersection

We construct some matroids that have a circuit and a cocircuit with infinite intersection.This answers a question of Bruhn, Diestel, Kriesell, Pendavingh and Wollan. It further shows that the axiom system for matroids proposed by Dress in 1986 does not axiomatize all infinite matroids.We show that one of the matroids we define is a thin sums matroid whose dual is not a thin sums matroid, answering a natural open problem in the theory of thin sums matroids.