  We generalize Baker-Bowler's theory of matroids over tracts to orthogonal
matroids, define orthogonal matroids with coefficients in tracts in terms of
Wick functions, orthogonal signatures, circuit sets, and orthogonal vector
sets, and establish basic properties on functoriality, duality, and minors. Our
cryptomorphic definitions of orthogonal matroids over tracts provide proofs of
several representation theorems for orthogonal matroids. In particular, we give
a new proof that an orthogonal matroid is regular if and only if it is
representable over $\mathbb{F}_2$ and $\mathbb{F}_3$, which was originally
shown by Geelen, and we prove that an orthogonal matroid is representable over
the sixth-root-of-unity partial field if and only if it is representable over
$\mathbb{F}_3$ and $\mathbb{F}_4$.
