Let $M$ be a real matrix of rank $r$ (and let us set $M=UV^T$, with $U,V^T\in\mathbb{R}^{n\times r}$, to fix the notation).

Let $|M|$ be the matrix obtained by taking the absolute value of each entry of $M$. Clearly $\operatorname{rk} |M|$ can be much smaller than $r$ --- take for instance Hadamard matrices.

However, what about the other direction? Is there a way to bound $\operatorname{rk} |M|$ from above in terms of $r$?

If I take random $U$ and $V$ with $n=200$, $r=2$, numerically $\operatorname{rk} |M|$ seems to be between 120 and 150 --- so definitely not as low as $r$ but also suspiciously far from being full-rank.