11. (20 points) Let complex numbers $z_{1}$ and $z_{2}$ satisfy $\operatorname{Re} z_{1}>0$, $\operatorname{Re} z_{2}>0$, and $\operatorname{Re} z_{1}^{2}=\operatorname{Re} z_{2}^{2}=2$. Find
(1) the minimum value of $\operatorname{Re} z_{1} z_{2}$;
(2) the minimum value of $\left|z_{1}+2\right|+\left|\overline{z_{2}}+2\right|-\left|\overline{z_{1}}-z_{2}\right|$.
11. (1) For $k=1,2$, let
$$
z_{k}=x_{k}+y_{k} \mathrm{i}\left(x_{k} \backslash y_{k} \in \mathbf{R}\right) \text {. }
$$

From the conditions, we know
$$
x_{k}=\operatorname{Re} z_{k}>0, x_{k}^{2}-y_{k}^{2}=\operatorname{Re} z_{k}^{2}=2 \text {. }
$$

Then $\operatorname{Re} z_{1} z_{2}=\operatorname{Re}\left(x_{1}+y_{1} \mathrm{i}\right)\left(x_{2}+y_{2} \mathrm{i}\right)$
$$
\begin{array}{l}
=x_{1} x_{2}-y_{1} y_{2}=\sqrt{\left(y_{1}^{2}+2\right)\left(y_{2}^{2}+2\right)}-y_{1} y_{2} \\
\geqslant\left(\left|y_{1} y_{2}\right|+2\right)-y_{1} y_{2} \geqslant 2 .
\end{array}
$$

Furthermore, when $z_{1}=z_{2}=\sqrt{2}$, $\operatorname{Re} z_{1} z_{2}=2$. This indicates that the minimum value of $\operatorname{Re} z_{1} z_{2}$ is 2.
(2) For $k=1,2$, map $z_{k}$ to the point $P_{k}\left(x_{k}, y_{k}\right)$ in the Cartesian coordinate system $x O y$, and let $P_{2}^{\prime}$ be the reflection of $P_{2}$ about the $x$-axis. Then the points $P_{1}$ and $P_{2}^{\prime}$ are both on the right branch of the hyperbola $C: x^{2}-y^{2}=2$.
Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C$, respectively.
It is easy to see that $F_{1}(-2,0), F_{2}(2,0)$.
By the definition of a hyperbola, we have
$$
\begin{array}{l}
\left|P_{1} F_{1}\right|=\left|P_{1} F_{2}\right|+2 \sqrt{2}, \\
\left|P_{2}^{\prime} F_{1}\right|=\left|P_{2}^{\prime} F_{2}\right|+2 \sqrt{2} . \\
\text { Therefore, }\left|z_{1}+2\right|+\left|\overline{z_{2}}+2\right|-\left|\overline{z_{1}}-z_{2}\right| \\
=\left|z_{1}+2\right|+\left|\overline{z_{2}}+2\right|-\left|z_{1}-\overline{z_{2}}\right| \\
=\left|P_{1} F_{1}\right|+\left|P_{2}^{\prime} F_{1}\right|-\left|P_{1} P_{2}^{\prime}\right| \\
=4 \sqrt{2}+\left|P_{1} F_{2}\right|+\left|P_{2}^{\prime} F_{2}\right|-\left|P_{1} P_{2}^{\prime}\right| \\
\geqslant 4 \sqrt{2} .
\end{array}
$$

The equality holds if and only if the point $F_{2}$ lies on the line segment $P_{1} P_{2}^{\prime}$ (for example, when $z_{1}=z_{2}=2+\sqrt{2} \mathrm{i}$, $F_{2}$ is exactly the midpoint of $P_{1} P_{2}^{\prime}$).

In summary, the minimum value of $\left|z_{1}+2\right|+\left|\overline{z_{2}}+2\right|-\left|\overline{z_{1}}-z_{2}\right|$ is $4 \sqrt{2}$.