5. The complex numbers $z_{1}, z_{2}$ satisfy
$$
\begin{array}{l}
\left|\left(\left|z_{1}\right| z_{1}\right)+\left(\left|z_{2}\right| z_{2}\right)\right| \\
=\left|\left(\left|z_{1}\right| z_{2}\right)+\left(z_{1}\left|z_{2}\right|\right)\right|,
\end{array}
$$

and $\left|z_{1}-\frac{1}{z_{2}}\right|+\left|z_{2}-\frac{1}{z_{1}}\right|$ has a maximum value and a minimum value whose sum is $4 \sqrt{3}$. Then $\left(\left|z_{1} \overline{z_{2}}\right|+\left|z_{2} \overline{z_{1}}\right|\right)^{2016}=$
5. $6^{2016}$ or $\left(\frac{2}{3}\right)^{2016}$.

From the given equation, we have $\left|z_{1}\right|=\left|z_{2}\right|$.
$$
\begin{array}{l}
\text { Also }|| z_{1}|-| \frac{1}{z_{2}}||+|| z_{2}|-| \frac{1}{z_{1}}|| \\
\leqslant\left|z_{1}-\frac{1}{z_{2}}\right|+\left|z_{2}-\frac{1}{z_{1}}\right| \\
\quad \leqslant\left|z_{1}\right|+\left|z_{2}\right|+\left|\frac{1}{z_{1}}\right|+\left|\frac{1}{z_{2}}\right|,
\end{array}
$$

Then $4\left|z_{1}\right|=4 \sqrt{3}\left(\left|z_{1}\right| \geqslant 1\right)$ or
$$
\begin{array}{l}
\frac{4}{\left|z_{1}\right|}=4 \sqrt{3}\left(\left|z_{1}\right| \leqslant 1\right) \\
\Rightarrow\left|z_{1}\right|=\sqrt{3} \text { or } \frac{\sqrt{3}}{3} \\
\Rightarrow\left(\left|z_{1} \overline{z_{2}}\right|+\left|z_{2} \overline{z_{1}}\right|\right)^{2016}=\left(2\left|z_{1}\right|^{2}\right)^{2016} \\
\quad=6^{2016} \text { or }\left(\frac{2}{3}\right)^{2016} .
\end{array}
$$