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 $\omega_2.$ Now working in $V[r][X],$
define a new sequence $(y_\alpha: \alpha<\omega),$ 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 $V_1=V[(y_\alpha: \alpha<\omega_1)]$ and $V_2=V_1[r].$  Note that:

1) $V_1\models GCH+\Diamond,$

2) $V_2=V[(y_\alpha: \alpha<\omega_1)][r]=V[r][X]\models GCH+\neg\Diamond,$

3) $V_2=V_1[r]$, for some real $r$.