= Solution
\b[The analogous flat-coordinate construction does not define a full $\mathrm{SL}_2(\mathbb R)$ action on arbitrary cubic differentials.] Away from zeros, a nonzero <holomorphic cubic differential> has <flat coordinates>
$$
w=\int c^{1/3},\qquad c=dw^3.
$$
Their changes of coordinate are $w_j=\zeta w_i+b_{ij}$ with $\zeta^3=1$. A branch change is therefore a rotation through $2\pi/3$, not merely a sign. Applying a real-linear map $A$ changes its linear part to $ARA^{-1}$, where $R$ is that rotation. In general this is not a <complex-linear map>, so the proposed changes of coordinate are not <holomorphic> and cannot define the required deformed <complex structure>.
For example, take $A=\operatorname{diag}(2,1/2)$ and $R=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$ with $\theta=2\pi/3$. Then
$$
ARA^{-1}=
\begin{pmatrix}\cos\theta&-4\sin\theta\\\frac14\sin\theta&\cos\theta\end{pmatrix},
$$
whose off-diagonal entries fail the condition for a <complex-linear map>. This dependence on the choice of cube-root coordinate is the obstruction even when considering descent from the locus $c=\omega^3$: the three possible roots need not lead to the same deformation of the cubic pair.
The real matrices preserving <orientation> that normalize the order-three rotations are precisely the matrices of <complex-linear maps>; intersecting with $\mathrm{SL}_2(\mathbb R)$ leaves $\mathrm{SO}(2)$. Indeed a nonreal rotation determines its <complex structure>, and conjugation to its inverse would reverse that structure's <orientation>. Thus there is a natural rotation action,
$$
\boxed{R_\theta\cdot(X,c)=(X,e^{3i\theta}c).}
$$
The conclusion concerns the geometric <group action> analogous to that for <translation surfaces> and <half-translation surfaces>; it does not rule out artificial group actions unrelated to these atlases.
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