= Solution
A <compatible almost complex structure> is a smooth endomorphism $J:TX\to TX$ satisfying $J^2=-I$, preserving $\omega$, and making $g_J(u,v)=\omega(u,Jv)$ a positive-definite <inner product>. Preservation and $J^2=-I$ make this <bilinear form> symmetric.
To construct one, choose any <Riemannian metric> $h$ and define $A$ by $h(Au,v)=\omega(u,v)$. The endomorphism $A$ is skew-adjoint and invertible. Consequently $-A^2$ is positive definite and self-adjoint. Let $B=(-A^2)^{1/2}$ be its positive square root and set
$$
\boxed{J=B^{-1}A.}
$$
The <spectral theorem> gives $B$ fibrewise, and the positive-square-root operation is smooth on positive-definite matrices, so this defines a smooth endomorphism. Since $A$ commutes with $B$, $J^2=-I$. Also $J^*=-J$, so $J^*J=I$. Commutation then gives
$$
\omega(Ju,Jv)=h(AJu,Jv)=h(Au,v)=\omega(u,v),\qquad
\omega(u,Ju)=h(Bu,u)>0\quad(u\ne0).
$$
This is the <metric construction of a compatible almost complex structure>, proving nonemptiness.
If we start with $h=g_J$ for an already compatible $J$, then $h(Ju,v)=\omega(u,v)$, so $A=J$, $B=I$, and the construction recovers $J$. For two compatible structures, linearly interpolate their metrics,
$$
h_t=(1-t)g_{J_0}+t g_{J_1},\qquad0\leq t\leq1,
$$
and apply the construction to $h_t$. This is a continuous smooth-in-$t$ path with endpoints $J_0,J_1$. Thus \b[the space is nonempty and path connected]. Interpolating instead to a single fixed auxiliary metric actually gives the <contractibility of compatible almost complex structures>.
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