The common principal solution of the equations $\sin \theta=-\frac{1}{2}$ and $\tan \theta=\frac{1}{\sqrt{3}}$ is

The common principal solution in $[0, 2\pi)$ satisfies both $\sin \theta = -\frac{1}{2}$ and $\tan \theta = \frac{1}{\sqrt{3}}$ simultaneously.

The sine equation has solutions $\frac{7\pi}{6}$ and $\frac{11\pi}{6}$, while the tangent equation yields $\frac{\pi}{6}$ and $\frac{7\pi}{6}$.

The intersection is $\frac{7\pi}{6}$, which corresponds to option C.

Final answer: $\boxed{\text{C}}$