= Solution
Fix ordered branch positions $P=(p_1,\ldots,p_r)$ and conjugacy classes $C_i$ in $G$. Form the generating product-one set
$$
\mathrm{Ni}(G;\mathbf C)=
\{(g_1,\ldots,g_r):g_i\in C_i,\ g_1\cdots g_r=1,
\ \langle g_1,\ldots,g_r\rangle=G\}.
$$
The <rigid ramification type> condition is that this set is nonempty and consists of one orbit under simultaneous inner conjugation by $G$. Rationality of classes is a further arithmetic condition and is not needed for this complex uniqueness statement.
Let $U=\mathbb P^1(\mathbb C)\setminus P$. Choose a base point and positively oriented peripheral loops $\gamma_i$ with sole relation $\gamma_1\cdots\gamma_r=1$. A connected Galois cover over $U$ gives a surjective <monodromy> map $\pi_1(U)\to G$, and changing the labeling of one fiber conjugates it. Its peripheral images are exactly a tuple in the set above.
For two extensions of the same rigid type, rigidity conjugates their tuples to agree. The loops generate $\pi_1(U)$, so their entire <monodromy> maps then agree. This constructs an isomorphism of the unramified covers directly: label fibers at the base point identically, and transport that matching along paths. Independence of path follows from equal <monodromy>. The resulting map is <holomorphic> because the cover maps are local biholomorphisms.
Near each missing point, the cover has local form $u=\pi^{e_i}$. The isomorphism therefore extends over the added points, equivalently by the removable-singularity theorem in these local parameters. Pullback of meromorphic functions gives an isomorphism of the function <fields> fixing $\mathbb C(t)$. Hence \b[there is at most one extension of the prescribed rigid type]. This is uniqueness for fixed branch positions and peripheral conventions, not uniqueness after arbitrarily moving the branch points.
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