Based on limited numerical evidence, I am inclined to suspect that
there is always zero of $\Re \zeta(1/2+it)$ between consecutive local 
extrema of $\Re \zeta(1/2+it)$
(and the same for $\Im \zeta(1/2+i t)$).

There are very short intervals having two zeros.

For Siegel $Z$ function, RH implies this for $t$ large enough.

>Is it true (maybe conditionally)?
>
>Counterexamples? (Please check for 2 zeros in a short interval).