The answer is yes, assuming the existence of $\aleph_2$-many measurable cardinals. To see this, assume $GCH+\Diamond$ holds and $S$ is a discrete set of measurable cardinals of size $\aleph_2.$
Step 1. Force with Prikry product forcing $P_S$ to change the cofinality of each element of $S$ into $\omega.$
Note that the extension is of the form $V[(x_\alpha: \alpha\in S)]$, where each $x_\alpha$ is an $\omega-$sequence cofinal in $\alpha.$
Step 2. Force with Jensen's coding theorem, to code everything into a real $r$, so that we have $V[(x_\alpha: \alpha\in S)][r]=V[r]$ (and in fact $=L[r]$).
Step 3. Force over $V[r]$, by a cardinal and $GCH$ preserving forcing iteration to force $\neg \Diamond$.
Note that the generic can be seen as a subset $X$ of $S.$ Now working in $V[r][X],$ define a new sequence $(y_\alpha: \alpha\in S),$ so that $y_\alpha=x_\alpha,$ if $\alpha\in X$ and $y_\alpha=x_\alpha\setminus\{ min(x_\alpha) \}$ if $\alpha\notin X.$
Then let $V_1=V[(y_\alpha: \alpha\in S)]$ and $V_2=V_1[r].$ Note that:
$V_1\models GCH+\Diamond,$
$V_2=V[(y_\alpha: \alpha\in S)][r]=V[r][X]\models GCH+\neg\Diamond,$
$V_2=V_1[r]$, for some real $r$.