  In 2004, Choe, Oxley, Sokal and Wagner established a tight connection between
matroids and multiaffine real stable polynomials. Recently, Branden used this
theory and a polynomial coming from the Vamos matroid to disprove the
generalized Lax conjecture. Here we present a 10-element extension of the Vamos
matroid and prove that its basis generating polynomial is real stable (i.e.
that the matroid has the half-plane property). We do this via large sums of
squares computations and a criterion for real stability given by Wagner and
Wei. Like the Vamos matroid, this matroid is not representable over any field
and no power of its basis generating polynomial can be written as the
determinant of a linear matrix with positive semidefinite Hermitian forms.
