$$\DeclareMathOperator{\Top}{Top}$$
$$\DeclareMathOperator{\co}{H}$$

Let $M$ be a closed orientable connected topological manifold of dimension $4k$ with $k > 1$. It is known ([David Frank, On the index of a tangent 2-field][1]) that if $M$ has a smooth structure, the necessary and sufficient condition for it to admit two linearly independent vector fields is the following:

> The Euler characteristic $\chi(M) = 0$ and the signature $\sigma(M)
 \equiv 0 \pmod{4}$.


Do the same two conditions characterize whether the tangent microbundle $\tau M$ of $M$ has a rank 2 trivial subbundle? 

The vanishing $\chi(M) = 0$ is evidently necessary, for instance by the Lefschetz fixed point theorem; it is also sufficient for $\tau M$ to have a rank **one** trivial subbundle by Corollary 1.5 in [Ronald J. Stern, On topological and piecewise linear vector fields][2]. 

Duane Randall has papers that work out the analogous characterization with non-orientable manifolds and manifolds of dimension $4k+1$, $4k+2$, $4k+3$, but they seem to leave the orientable 4k case unsettled in general.
<br>
[Duane Randall, CAT 2-fields on nonorientable CAT manifolds][3]
<br>
[Duane Randall, On indices of tangent fields with finite singularities][4]
<br>
[Duane Randall, On 4-manifolds and span-related numbers for CAT manifolds][5]


Let me outline the formal part of (what seems to be) the right setup. By microbundle representation theorems we can write $\tau \colon M \rightarrow B\Top(4k)$ which we want to lift along a fibration
$$V_{4k,2}^{\Top} := \Top(4k) \,/\, \Top(4k,2) \rightarrow B\Top(4k,2) \rightarrow B\Top(4k) \, .$$
The primary obstruction should be the Bockstein of the $(4k-2)$nd Stiefel-Whitney class and vanish, similar to the smooth case. The next and final obstruction class will lie inside
$$\co^{4k}\left(M;\pi_{4k-1}(V_{4k,2}^{\Top})\right) \cong \pi_{4k-1}(V_{4k,2}^{\Top}) \cong \pi_{4k-1}(V_{4k,2}) \cong \mathbb{Z} \oplus \mathbb{Z}/2$$
where the second isomorphism is by Theorem 2.5 of Stern's paper. The problem then becomes showing that this class always corresponds to a pair of the form
$$\left(\chi(M), \,\frac{\sigma(M) \pm \chi(M)}{2} \!\!\!\mod{\!2}\right) \in \mathbb{Z} \oplus \mathbb{Z}/2 \, .$$


  [1]: https://www.sciencedirect.com/science/article/pii/0040938372900110
  [2]: https://www.sciencedirect.com/science/article/pii/0040938375900075
  [3]: https://academic.oup.com/qjmath/article-abstract/38/3/355/1533968
  [4]: https://link.springer.com/chapter/10.1007/BFb0081477
  [5]: https://link.springer.com/article/10.1007/BF02567932