case, the manifold $M^4$ is not anti-self-dual and an almost complex structure on ${{\cal T}}(M^4)$ is not integrable.[^4] Here, we show that this generalized twistor correspondence holds for the case of an arbitrary oriented 4-manifold $M^4$. Namely, we describe a correspondence between Hermitian vector bundles $E$ with ASD connections $A$ on an oriented 4-manifold $M^4$ and pseudo-holomorphic vector bundles $\hat E$ over an almost complex twistor space ${{\cal T}}(M^4)$.

[**Almost complex structure on ${{\cal T}}(M^4)$**]{}. We fix an open subset[^5] ${{\cal U}}$ of $M^4$ with coordinates $\{x^{\m}\}$, $\m =1,...,4$, and an open subset $U={\mathbb C}P^1\setminus\{\infty\}$ of ${\mathbb C}P^1$ with a local complex coordinate $\zeta$. Then ${{\cal U}}\times U$ is an open subset of ${{\cal T}}(M^4)$. Note that over ${{\cal U}}$ there exists a section $J=(J^\n_\m )$ of the bundle (\[3.2\]) (a local almost complex structure) and this allows to introduce forms of type $(p,q)$ w.r.t. $J$. Globally such an almost complex structure $J$ on $M^4$ may not exist but this is not necessary for all twistor constructions.

Let $\{\vt^\m\}$ represents some orthonormal coframe on ${{\cal U}}\subset M^4$, i.e. $\diff s^2=\de_{\m\n}\vt^\m\vt^\n$. Using the canonical form of a local almost complex structure $J$, we introduce forms $$\label{3.4}
\th^1:=\vt^1+\im\vt^2\ ,\quad\th^2:=\vt^3+\im\vt^4\ ,\quad\th^{\1}:=\vt^1-
\im\vt^2\und\th^{\2}:=\vt^3-\im\vt^4\ ,$$ 