Here are a couple of contour plots of
$$\Re(\zeta(\alpha+i \beta))=\Im(\zeta(\alpha+i \beta))\quad\tag{blue curve}$$
$$\Re(\zeta(\alpha+i \beta))=0\quad\tag{orange curve}$$
$$\Im(\zeta(\alpha+i \beta))=0\quad\tag{green curve}$$
where the red discrete points are the non-trivial zeta zeros.
Note $\Re(\zeta(\alpha+i \beta))=0$ (orange curve) and $\Im(\zeta(\alpha+i \beta))=0$ (green curve) are both symmetric about the real axis whereas $\Re(\zeta(\alpha+i \beta))=\Im(\zeta(\alpha+i \beta))$ (blue curve) is not.
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Here's a region plot of
$$\Re(\zeta(\alpha+i \beta))>\Im(\zeta(\alpha +i \beta))\quad\tag{blue region}$$
$$\Re(\zeta (\alpha +i \beta ))<\Im(\zeta (\alpha +i \beta ))\quad\tag{orange region}$$
where the red discrete points are the non-trivial zeta zeros.
Note the boundaries where these two regions meet correspond to the blue curves in the contour plots above.
