A Note on Schoen's Calabi--Yau Threefolds

We study the topology of a real Lagrangian in Schoen’s Calabi–Yau threefold $X$ and compute its mod $2$ cohomology using two methods; first via a concrete Mayer–Vietoris calculation, then by an exact sequence relating the mod $2$ cohomology of the real Lagrangian to the cohomology of $X$. We conclude that these two methods agree. This in particular corrects a previous computation made by Casta~{n}o-Bernard–Matessi.