= Solution
An <almost complex structure> is a smooth real <vector bundle endomorphism> $J:TM\to TM$ satisfying $J^2=-I$. In particular, $M$ has even real dimension $2n$. Extend $J$ complex-linearly to $T_{\mathbb C}M=TM\otimes_{\mathbb R}\mathbb C$. Its minimal polynomial divides $(u-i)(u+i)$, whose roots are distinct, so the <type decomposition of the complexified tangent bundle> is
$$
T_{\mathbb C}M=T^{1,0}M\oplus T^{0,1}M,\qquad
T^{1,0}M=\ker(J-iI),\quad T^{0,1}M=\ker(J+iI).
$$
The smooth projections are $(I-iJ)/2$ and $(I+iJ)/2$. <Complex conjugation> exchanges the two rank-$n$ <eigenbundles>. A $(1,0)$-covector annihilates $T^{0,1}M$ and a $(0,1)$-covector annihilates $T^{1,0}M$. Thus the <differential forms of type (p, q)> are the smooth sections of
$$
\Lambda^{p,q}T^*M=\Lambda^p(T^{1,0}M)^*\otimes\Lambda^q(T^{0,1}M)^*,
\qquad
\Lambda^rT^*_{\mathbb C}M=\bigoplus_{p+q=r}\Lambda^{p,q}T^*M.
$$
Here the tensor product is identified with its image under the <wedge product of differential forms>. For a real tangent vector $v$, the vectors $v-iJv$ and $v+iJv$ exhibit the two types directly.
We now construct the <almost complex structure from a decomposable volume form>. Locally write $\Omega=\theta_1\wedge\cdots\wedge\theta_n$. The nonvanishing of $\Omega\wedge\overline\Omega$ says that the $2n$ covectors $\theta_1,\ldots,\theta_n,\overline\theta_1,\ldots,\overline\theta_n$ form a complex coframe. In particular, the span of the $\theta_j$ has the intrinsic description
$$
W_x=\{\alpha\in T_x^*M\otimes\mathbb C:\alpha\wedge\Omega_x=0\}.
$$
Indeed, expansion in that coframe shows that every coefficient of a conjugate factor must vanish if $\alpha\wedge\Omega_x=0$. Therefore this span is independent of the chosen local factorization. The local coframes show that $W$ is a smooth rank-$n$ <vector subbundle>, and
$$
T^*M\otimes\mathbb C=W\oplus\overline W.
$$
Define $J_\Omega^*$ to be multiplication by $i$ on $W$ and by $-i$ on $\overline W$. It commutes with <complex conjugation>, so is the complexification of a real smooth <bundle endomorphism>, whose dual defines $J_\Omega$ on $TM$. It has square $-I$, and its $(1,0)$-covectors are exactly $W$. Conversely, any such <almost complex structure> must have this same action on the entire coframe. Thus \b[$J_\Omega$ exists globally and is unique], and every factor in every allowed decomposition is a $(1,0)$-form.
Finally suppose $d\Omega=0$. Since $\theta_j\wedge\Omega=0$, the <exterior derivative> product rule gives
$$
0=d(\theta_j\wedge\Omega)=d\theta_j\wedge\Omega-\theta_j\wedge d\Omega
=d\theta_j\wedge\Omega.
$$
Decompose $d\theta_j$ into types $(2,0)$, $(1,1)$ and $(0,2)$ using $J_\Omega$. The first two wedge products with the $(n,0)$-form $\Omega$ vanish by dimension. On $(0,2)$-forms, wedging with $\Omega$ is injective: the monomials $\Omega\wedge\overline\theta_a\wedge\overline\theta_b$ are linearly independent. For $n=1$ the $(0,2)$ space is zero, so the statement still holds. Hence every $d\theta_j$ has no $(0,2)$ component. For an arbitrary $(1,0)$-form $\alpha=\sum_j a_j\theta_j$, the extra terms $da_j\wedge\theta_j$ likewise have only types $(2,0)$ and $(1,1)$. The <bracket and differential-form criteria for integrability> proved below now give
$$
\boxed{d\Omega=0\ \Longrightarrow\ J_\Omega\text{ is integrable}.}
$$
Local decomposability has been used essentially; the nonvanishing top-degree product alone does not provide the required coframe.
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