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We prove that a natural signed count of the $2$-torsion points on a realprincipally polarized abelian variety $A$ always equals to $2^{g}$ where $g$ isthe dimension of $A$. When $A$ is the Jacobian of a real curve we derive signedcounts of real odd theta characteristics. These can be interpreted in terms ofthe extrinsic geometry of contact hyperplanes to the canonical embedding of thecurve. We also formulate a conjectural generalization to arbitrary fields interms of $\mathbb{A}^1$-enumerative geometry.

A signed count of 2-torsion points on real abelian varieties