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Fix ordered branch positions $P=(p_1,\dots ,p_r)$ and conjugacy classes $C_i$ in $G$. Form the generating product-one setThe 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 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.