We prove that a natural signed count of the 2-torsion points on a real, principally polarized Abelian variety A always equals to 2g, where g is the dimension of A. When A is the Jacobian of a real curve, we derive signed counts of real odd theta characteristics. These can be interpreted in terms of the extrinsic geometry of contact hyperplanes to the canonical embedding of the curve. We also formulate a conjectural generalization to arbitrary fields in terms of the A1-enumerative geometry.