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A surjective inverse-polynomial germ projection without even covering comes from the inverse of $P(w)=w^3-3w$ over $G=\mathbb{C}$. Consider $R=\mathbb{C}\setminus \{-1,1\}$, with $\pi (w)=P(w)$. Since $P^{\prime }(w)=3(w^2-1)\ne 0$ on $R$, each $w$ defines a holomorphic inverse germ of $P$ at $P(w)$. This identifies $R$ with a component of the space of germs of holomorphic functions: inverse germs continue along paths in $R$, and the holomorphic identity $P(f(z))=z$ persists under continuation. A germ with value $\pm 1$ cannot exist, because differentiating that identity would give $P^{\prime }(\pm 1)f^{\prime }(z)=1$. Since $R$ is connected, all its inverse germs belong to one complete analytic function.

The projection is surjective. A generic value has three distinct regular preimages; the critical values $2,-2$ each retain a simple regular preimage, respectively $2,-2$, even after their double critical preimages $-1,1$ are removed. Thus fibre size drops from $3$ to $1$ at these values, making an evenly covered neighbourhood impossible. This is surjective but not a regular topological covering.

If “regular” is specifically used for deck transitivity among genuine coverings, there is also a genuine nonnormal example. Remove the critical values from the base and all their preimages from the source:This is a connected three-sheeted analytic covering. A small loop about each critical value swaps the two roots meeting there. The two swaps are distinct and generate $S_3$ (connectedness also forces transitive monodromy). The stabilizer of one sheet is an order-$2$ subgroup, which is not normal in $S_3$. The classification of connected covering spaces therefore shows this covering is not regular in the deck-transitive sense. Either convention has a fully explicit example.