Based on limited numerical evidence, I suspect this conjecture.
Conjecture: Fix $ 0 \le \sigma \le \frac12$ and let $t > 0$. Between consecutive local extrema of $\Re \zeta(\sigma+i t)$ (resp. $\Im \zeta(\sigma+ it)$), there is always a zero of $\Re \zeta(\sigma+i t)$ (resp. $\Im \zeta(\sigma+ it)$).
Verification for several random $\sigma$ and $ 0 < t < 30000$ and for a few random intervals didn't show any counterexamples.
For $\sigma > \frac12$ it is false and on the other hand this appears counterintuitive to me.
Counterexamples? (please check for closely spaced zeros that might look like a single local minimum on a large plot).
Does this contradict something?
Even if it is true, a conditional proof probably will be hard yet welcome.
For Siegel $Z$ function on the critical line RH implies this for $t$ large enough.
Maybe can be generalized to $\sigma \le \frac12$.
Plot of a random interval: