Let $M$ be a manifold. A complex structure on $M$ is an endomorphism $J \in \text{End}(TM)$ such that $J^2 = -\text{id}$ together with the vanishing of the Nijenhuis tensor. If $J$ is real-analytic, the Frobenius theorem implies that the data $(M,J)$ defines a complex manifold, i.e., there exist local holomorphic coordinates on $M$. If $J$ is smooth, then $(M,J)$ is a complex manifold by the Newlander-Nirenberg theorem. If $J$ is only assumed to be $C^1$, does the vanishing of the Nijenhuis tensor imply that $(M,J)$ is a complex manifold?

cannotassume M is endowed with a complex structure (which I am taking to mean what differential geometers usually call an "almost complex" structure), since in those cases it has none. $\endgroup$