Does the locally-ringed spaces viewpoint on topology actually do what we want?

Question

There's a post here about how we know that the morphisms of smooth manifolds as locally-ringed spaces are the same as the morphisms of smooth manifolds as charts and atlases. I didn't understand the argument, but it was mentioned that the argument breaks down for fields other than $\mathbb{R}$. Does this mean that attempts to take a locally-ringed spaces perspective on various kinds of manifolds over anything other than $\mathbb{R}$ will tend to either fail, or at least have some surprising and perhaps undesirable pathologies due to the morphisms being "too general"?