  The quintic threefold $X$ is the most studied Calabi-Yau $3$-fold in the
mathematics literature. In this paper, using \v{C}ech-to-derived spectral
sequences, we investigate the mod $2$ and integral cohomology groups of a real
Lagrangian $\breve{L}_{\mathbb{R}}$, obtained as the fixed locus of an
anti-symplectic involution in the mirror to $X$. We show that
$\breve{L}_{\mathbb{R}}$ is the disjoint union of a $3$-sphere and a rational
homology sphere. Analysing the mod $2$ cohomology further, we deduce a
correspondence between the mod $2$ Betti numbers of $\breve{L}_{\mathbb{R}}$
and certain counts of integral points on the base of a singular torus fibration
on $X$. By work of Batyrev, this identifies the mod $2$ Betti numbers of
$\breve{L}_{\mathbb{R}}$ with certain Hodge numbers of $X$. Furthermore, we
show that the integral cohomology groups
$H^j(\breve{L}_{\mathbb{R}},\mathbb{Z})$ of $\breve{L}_{\mathbb{R}}$ are
$2$-primary for $j \neq 0,3$; we conjecture that this holds in much greater
generality.
