Prikry-Silver forcing adds a minimal real: that is, if $g$ is Prikry-Silver generic over $V$ and $h$ is any real in $V[g] \setminus V$, then $V[g] = V[h]$.

Interestingly, while Prikry-Silver forcing is minimal for reals, it is not fully minimal (like Sacks forcing). If $g$ is Prikry-Silver generic over $V$, then there is a set $F \in V[g] \setminus V$ such that $V[F] \neq V[g]$. (Specifically, this is true when $F$ is the filter generated by all complements of domains of conditions in the generic; or, equivalently, if $g$ is the Prikry-Silver generic real, then $F$ is the filter generated by $\{ A \in V \cap \mathcal P(\omega) : g \restriction A \notin V \}$. Then $F$ is a new non-principal ultrafilter on $\omega$, and $V[F]$ contains no new reals.)

**Edit:**
The minimality-for-reals of Prikry-Silver forcing can be seen as a special case of the following theorem of Grigorieff:

If $\mathcal U$ is a selective ultrafilter, then (what's now called) the Grigorieff forcing on $\mathcal U$ adds a minimal real. The conditions in this forcing are the partial functions $A \rightarrow 2$ where $A \notin \mathcal U$, and the ordering is the natural one (extension).

The reason this implies Prikry-Silver forcing adds a minimal real (but is not fully minimal) is that Prikry-Silver forcing can be factored as a $2$-step iteration: first force with $\mathcal P(\omega)/\mathrm{fin}$ to add a selective ultrafilter, and then force with the Grigorieff forcing on that ultrafilter. The first step adds no new reals, and the second step adds a minimal real by the theorem quoted above.

(All of this stuff can be found in this paper of Grigorieff. This old MO thread is also relevant.)