  We study the mod $2$ cohomology of real Calabi-Yau threefolds given by real
structures which preserve the torus fibrations constructed by Gross. We extend
the results of Casta\~no-Bernard-Matessi and Arguz-Prince to the case of real
structures twisted by a Lagrangian section. In particular we find exact
sequences linking the cohomology of the real Calabi-Yau with the cohomology of
the complex one. Applying SYZ mirror symmetry, we show that the connecting
homomorphism is determined by a ``twisted squaring of divisors'' in the mirror
Calabi-Yau, i.e. by $D \mapsto D^2 + DL$ where $D$ is a divisor in the mirror
and $L$ is the divisor mirror to the twisting section. We use this to find an
example of a connected $(M-2)$-real quintic threefold.
