Why does the log-rank conjecture use rank over the reals?
The conjecture fails over $\mathbb{F}_2$. Look at $M(x, y) = \langle x, y \rangle \bmod 2$, and $x, y \in \{0, 1\}^n$. The communication complexity is $\Omega(n)$, but the rank of $M$ over $\mathbb{F}_2$ is $n$, by the linearity of inner product.