5. Given the complex number $z$ satisfies $|z|=1$, and $\operatorname{Re} \frac{z+1}{\bar{z}+1}=\frac{1}{3}$, then $\operatorname{Re} \frac{z}{\bar{z}}$ is $\qquad$ .
Given $|z|=1 \Rightarrow \bar{z}=\frac{1}{z}$, then $\frac{z+1}{\bar{z}+1}=\frac{z+1}{\frac{1}{z}+1}=z, \frac{z}{\bar{z}}=z^{2}$, let $z=\cos \theta+\mathrm{i} \sin \theta$, thus $\cos \theta=\frac{1}{3} \Rightarrow \cos 2 \theta=2 \cos ^{2} \theta-1=-\frac{7}{9}$. Therefore, the value of $\operatorname{Re} \frac{z}{\bar{z}}$ is $-\frac{7}{9}$.