On real Calabi-Yau threefolds twisted by a section

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 Castano-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 --> 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.