Question
Does there exist a symplectic form $\omega$ on $\mathbb{C}P^3$ with first Chern class $c_1(\mathbb{C} P^3, \omega) = 0$?
Context
Let $(M,\omega)$ be a symplectic manifold. An almost complex structure $J$ on $M$ is compatible with $\omega$ if the bilinear form $(u,v) \mapsto \omega(u, Jv)$ is an inner product. The space of compatible structures is non-empty and contractible. This allows us to define the Chern classes $c_k(M,\omega)$ as $c_k(TM, J)$, where $J$ is any compatible almost complex structure on $M$.
It is well known that the Fubini-Study form $\omega_{\operatorname{FS}}$ on $\mathbb{C} P^3$ is compatible with the standard complex structure, making $\mathbb{C} P^3$ into a Kähler manifold (and more generally the same is true for $\mathbb{C}P^n$). Note that in this case we have $c_1(\mathbb{C}P^3, \omega_{\operatorname{FS}}) = \pm 4 x$, where $x \in H^2(\mathbb{C}P^3; \mathbb{Z})$ is a generator.
On the other hand, $\mathbb{C} P^3$ has many almost complex structures that are not isomorphic to the standard one. For example, one finds the following result in [1] (see Theorem 3.2 therein).
For every $q \in \mathbb{Z}$, there is an almost complex structure $J$ on $\mathbb{C} P^3$ with $c_1(\mathbb{C} P^3, J) = 2 q x$, where $x \in H^2(\mathbb{C}P^3; \mathbb{Z})$ is a choice of generator.
The above claim can be proved using obstruction theory. On the other hand, I do not know of an explicit construction of the non-standard almost complex structures.
This leads to the question: are any of the non-standard almost complex structures on $\mathbb{C}P^3$ compatible with a symplectic form? The question stated above is concerned with the special case $c_1 = 0$.
[1] Thomas, E. Complex structures on real vector bundles. Amer. J. Math.89(1967), 887–908. MR0220310