= Solution
On <complexified Minkowski spacetime>, use the <twistor incidence relation>
$$
\boxed{\omega^A=ix^{AA'}\pi_{A'},\qquad \pi\ne0.}
$$
The harmless factor $i$ is a conventional choice; omitting it corresponds to rescaling the complexified coordinates. For fixed $x$, projectivizing the nonzero <two-component spinors> $\pi$ gives a <twistor line> $L_x\simeq\mathbb{CP}^1$. The two stated patches cover all these incidence lines; they describe the affine-space-time twistor domain with the line $\pi=0$ removed from full <projective twistor space>.
Since $E|_{L_x}$ is holomorphically trivial, choose a global <bundle frame> on that line. Comparing its columns with the <bundle frames> in the two patch trivializations gives invertible <matrices> $H(x,\pi)$ and $\widetilde H(x,\pi)$, holomorphic on their respective patches, with
$$
\boxed{F(ix\pi,\pi)=\widetilde H(x,\pi)H(x,\pi)^{-1}.}
$$
This is the <holomorphic splitting of a twistor patching matrix>. It follows directly from existence of the global <bundle frame>, rather than from an arbitrary factorization of an arbitrary <matrix>: nontrivial line bundles would obstruct such a nonsingular splitting.
The factors can be chosen holomorphically in $x$ near any given $x_0$. An explicit local argument is useful. Normalize the known splitting at $x_0$ and set $Q(x,\lambda)=\widetilde H(x_0,\lambda)^{-1}F(x,\lambda)H(x_0,\lambda)$ on a fixed circular contour in the overlap. It is close to the identity when $x$ is close to $x_0$. Let $P_+$ select strictly positive Laurent powers, and seek $q=1+$ positive powers satisfying
$$
q=1-P_+[(Q-1)q].
$$
In a holomorphic contour-function norm the operator has norm below one for sufficiently small variation, so its <Neumann series> converges and is holomorphic in $x$. Then $Qq=\widetilde q$ has only nonpositive powers and is holomorphic on the exterior patch including infinity. Both factors are close to the identity and invertible. They extend over their full patches through $Q$ and its inverse. Taking $H=H(x_0)q$ and $\widetilde H=\widetilde H(x_0)\widetilde q$ gives the required local family. This is also the elementary <Laurent series> reason that $H^1(\mathbb{CP}^1,\mathcal O)=0$ leaves no infinitesimal obstruction.
Raise primed <two-component spinor> indices with the antisymmetric <two-component spinor> form and define $V_A=\pi^{A'}\partial_{AA'}$, holding $\pi$ fixed. Incidence gives
$$
V_A\omega^B=i\delta_A^B\pi^{A'}\pi_{A'}=0,\qquad V_A\pi_{B'}=0.
$$
Thus $V_AF=0$ for a pulled-back <function> depending only on twistor coordinates. Differentiating the splitting and multiplying by the inverse factors gives
$$
H^{-1}V_AH=\widetilde H^{-1}V_A\widetilde H.
$$
The left expression is holomorphic on one <two-component spinor> patch, the right on the other, and each is homogeneous of degree one in $\pi$. They therefore glue to a global matrix-valued section of $\mathcal O(1)$ on the line.
Such a section is linear in its homogeneous coordinates. To see this explicitly, set $\lambda=\pi_{0'}/\pi_{1'}$ on $U$. The normalized expression is an entire <matrix> <function> of $\lambda$; on the other patch its quotient by $\lambda$ is holomorphic at infinity. Hence it has at most linear growth. <Cauchy estimates> force all its <derivatives> of order at least two to vanish. Entry by entry it is $a_A(x)+\lambda b_A(x)$, or invariantly
$$
\boxed{\pi^{A'}\Phi_{AA'}(x)=H^{-1}\pi^{A'}\partial_{AA'}H.}
$$
The coefficients are holomorphic in $x$. This proves existence of the desired space-time one-form, with no remaining dependence on $\pi$ in its coefficients.
A second splitting must differ by right multiplication $H\mapsto Hg(x)$ and $\widetilde H\mapsto\widetilde Hg(x)$. Indeed the comparison <matrix> is holomorphic on both patches of a compact <complex projective line> and therefore constant along the line. The resulting transformation is
$$
\Phi\longmapsto g^{-1}\Phi g+g^{-1}dg,
$$
the <gauge connection> transformation law. Local coefficient <matrices> consequently glue to a holomorphic <gauge connection> on the bundle whose fiber at $x$ is the space of <holomorphic sections> of $E|_{L_x}$. If a single global matrix-valued one-form is wanted on all $M_{\mathbb C}\simeq\mathbb C^4$, choose a basis at the origin and parallel-transport it along the straight radial paths to each $x$. Linear transport exists on every finite path and varies holomorphically with $x$; this gives a global holomorphic <bundle frame>. The assertion does not require <flat connection> in all four directions, only a fixed choice of paths for a trivialization. On more general space-time domains the intrinsic object is a <gauge connection> with these gauge-related local one-forms.
Finally put $D_{AA'}=\partial_{AA'}+\Phi_{AA'}$ and $\mathcal D_A(\pi)=\pi^{A'}D_{AA'}$. From the defining identity,
$$
\mathcal D_A(\pi)H^{-1}
=-H^{-1}(V_AH)H^{-1}+(H^{-1}V_AH)H^{-1}=0.
$$
Thus the columns of $H^{-1}$ are an invertible parallel <bundle frame> in the two <alpha-plane> directions. Their <matrix commutator> must annihilate that <bundle frame>, so
$$
\pi^{A'}\pi^{B'}F_{AA'BB'}=0\qquad\text{for every }\pi.
$$
Use the <spinor decomposition of gauge curvature>
$$
F_{AA'BB'}=\epsilon_{AB}\varphi_{A'B'}+\epsilon_{A'B'}\varphi_{AB},
$$
where both <gauge curvature> <two-component spinors> are symmetric. The second term vanishes under contraction with $\pi^{A'}\pi^{B'}$. The first is a quadratic polynomial in the two <two-component spinor> components; vanishing for every <two-component spinor> forces each of its coefficients to vanish. Therefore
$$
\boxed{\varphi_{A'B'}=0,}
$$
which is precisely the <ASDYM> equation in the convention where the primed symmetric <two-component spinor> is the self-dual part. Equivalently, the <gauge connection> is flat when restricted to every <alpha-plane>. This completes the forward <Penrose-Ward correspondence>, including both the existence of the <gauge connection> and its <anti-self-duality of gauge curvature>.
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