= Solution
With <left Grassmann derivatives>, differentiating $\theta\sigma^\mu\bar\theta$ with respect to $\bar\theta$ introduces a minus sign. Consequently
$$
\bar D_{\dot\alpha}y^\mu=-(-i\theta^\beta\sigma^\mu_{\beta\dot\alpha})-i\theta^\beta\sigma^\mu_{\beta\dot\alpha}=0,\qquad y^\mu=x^\mu+i\theta\sigma^\mu\bar\theta.
$$
At fixed $y$, the chirality condition becomes independence of $\bar\theta$. There are only two independent <Grassmann variables> $\theta^\alpha$, so the <chiral-superfield component expansion> terminates:
$$
\boxed{\Phi(y,\theta)=\phi(y)+\sqrt2\,\theta^\alpha\psi_\alpha(y)+\theta^2F(y).}
$$
Here $\phi$ is a <complex scalar field>, $\psi$ a <Weyl spinor>, and $F$ a complex <auxiliary field>. Its four real off-shell bosonic components, two in $\phi$ and two in $F$, match the four real off-shell fermionic components. Equivalently, in ordinary coordinates the complete expansion is fixed without ambiguous contraction signs by
$$
\Phi(x,\theta,\bar\theta)=e^{i\theta\sigma^\mu\bar\theta\partial_\mu}\bigl[\phi(x)+\sqrt2\theta\psi(x)+\theta^2F(x)\bigr].
$$
The exponential terminates because of the <Grassmann algebra>.
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