  Let $r \leqslant n$ be nonnegative integers, and let $N = \binom{n}{r} - 1$.
For a matroid $M$ of rank $r$ on the finite set $E = [n]$ and a partial field
$k$ in the sense of Semple--Whittle, it is known that the following are
equivalent: (a) $M$ is representable over $k$; (b) there is a point $p = (p_J)
\in {\bf P}^N(k)$ with support $M$ (meaning that $\text{Supp}(p) := \{J \in
\binom{E}{r} \; \vert \; p_J \ne 0\}$ of $p$ is the set of bases of $M$)
satisfying the Grassmann-Pl\"ucker equations; and (c) there is a point $p =
(p_J) \in {\bf P}^N(k)$ with support $M$ satisfying just the 3-term
Grassmann-Pl\"ucker equations. Moreover, by a theorem of P. Nelson, almost all
matroids (meaning asymptotically 100%) are not representable over any partial
field. We prove analogues of these facts for Lagrangian orthogonal matroids in
the sense of Gelfand-Serganova, which are equivalent to even Delta-matroids in
the sense of Bouchet.
