Let $V$ be a real vector bundle on a space $X$, perhaps the tangent bundle of a smooth compact manifold. I'm interested in understanding the obstructions to $V$ admitting a stable complex structure, and also understanding how much this differs from the unstable story (if we ask for $V$ to admit a complex structure). Here are some things I know.

On the one hand, there are necessary conditions coming from characteristic classes. Namely, the odd Stiefel-Whitney classes $w_{2k+1}(V) \in H^{2k+1}(X, \mathbb{F}_2)$ must vanish, and the even Stiefel-Whitney classes must be in the image of the reduction map $H^{2k}(X, \mathbb{Z}) \to H^{2k}(X, \mathbb{F}_2)$, or equivalently the odd integral Stiefel-Whitney classes $\beta w_{2k}(V) = W_{2k+1}(V) \in H^{2k+1}(X, \mathbb{Z})$ must vanish, where $\beta$ is the Bockstein map $H^{2k}(X, \mathbb{F}_2) \to H^{2k+1}(X, \mathbb{Z})$. I don't know if this is sufficient in general. It certainly doesn't suffice for complex, rather than stable complex, structures, since even spheres have no odd cohomology but already $S^4$ doesn't have an almost complex structure.

On the other hand, by obstruction theory we need to look at the fibration

$$O/U \to BU \to BO$$

since lifting the stable classifying map $X \to BO$ of $V$ to a classifying map $X \to BU$ is equivalent to finding sections of an associated bundle with fibers $O/U$. The obstructions to doing this are cohomology classes in $H^{i+1}(X, \pi_i(O/U))$. Now, Bott periodicity implies that $O/U \cong \Omega O$, so its homotopy groups are known: they are periodic with period $8$ and the nontrivial ones are

$$\pi_{8k}(O/U) \cong \pi_{8k+7}(O/U) \cong \mathbb{Z}_2$$

and

$$\pi_{8k+2}(O/U) \cong \pi_{8k+6}(O/U) \cong \mathbb{Z}.$$

So there are obstructions living in $H^{8k+1}(X, \mathbb{Z}_2), H^{8k+8}(X, \mathbb{Z}_2), H^{8k+3}(X, \mathbb{Z})$, and $H^{8k+7}(X, \mathbb{Z})$. Three of these live in the same groups as the characteristic classes above so one might hope that they are in fact the same obstructions, but the obstructions living in $H^{8k+8}(X, \mathbb{Z}_2)$ don't match up. What's up with those?

For the unstable picture, when $\dim V = 2n$ we need to look at the fibration

$$O(2n)/U(n) \to BU(n) \to BO(2n).$$

There are induced maps $O(2n)/U(n) \to O(2n+2)/U(n+1)$ which induce isomorphisms on $\pi_k$ for (if I've calculated this correctly) $k \le 2n - 2$, so for example if we only cared about tangent bundles the stable and unstable stories almost match up except for the possibility of a mismatch involving classes in $H^{2n}(X, \pi_{2n-1}(O(2n)/U(n))$. For example, when $n = 2$ we have $O(4)/U(2) \cong S^2 \sqcup S^2$ and $\pi_0, \pi_1, \pi_2$ match up with the stable values above but $\pi_3$ is $\mathbb{Z}$ instead of being trivial.

Here are some questions I have.

How does the Stiefel-Whitney class story match up to the obstruction theory story? To what extent can we identify the obstructions involved in the two stories with each other? And how different are the stable and unstable obstructions?

This question is closely related but I don't think it completely answers my questions.