  We give a topological description of the two-row Springer fiber over the real
numbers. We show its cohomology ring coincides with the oddification of the
cohomology ring of the complex Springer fiber introduced by Lauda-Russell. We
also realize Ozsv\'ath-Rasmussen-Szab\'o odd TQFT from pullbacks and
exceptional pushforwards along inclusion and projection maps between hypertori.
Using these results, we construct the odd arc algebra as a convolution algebra
over components of the real Springer fiber, giving an odd analogue of a
construction of Stroppel-Webster.
