Why does every noncompact orientable surface have a complex structure?

Given a Riemannian metric, giving a compatible almost complex structure is equivalent to giving a reduction of the holonomy group from $O(n)$ to $U(n)$. An orientable surface must have holonomy group in $SO(2)$, which is just $U(1)$, so the parallel transport automatically preserves a compatible almost complex structure chosen at some point. And in two dimensions, the Nijenhuis tensor automatically vanishes so an almost complex structure is complex. None of this called on compactness.

The countable connect sum of tori is an example of a noncompact orientable surface that doesn't embed in the plane, unless I'm being silly.

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