Consider a structure $K$ comprised of four open quadrants, which form a disconnected Riemannian manifold. If we add in the axes, assume that the metric blows up along the axes. Call $L$ the structure with the axes included.
A more general description of $L$ is a manifold with singular curves in which the metric blows up along.
Is there a name for such a structure $L$? Does $L$ arise in any natural way?
An example to show it's possible to construct $L$: Transform $\Bbb R^2,$ with a diffeomorphism $f(x,y)=(e^x,e^y)$ and compute the metric in the new coordinates. One gets $ds^2=\frac{dx^2}{x^2}+\frac{dy^2}{y^2}.$ Then one can glue together four copies of this manifold in such a way that the metric blows up along the axes because of the division by $0.$
$L$ is not a Riemannian manifold because the distance between any two points must be finite.
If I understand correctly, you're looking for a natural way to define a complete Riemannian metric on an open subspace of your space, obtained by taking the complement of your singular lines. As far as I'm aware, there is no natural way to obtain such a metric in general, even if a previous metric is provided.
Here's an approach that might be of interest to you though: Given a complete Riemannian manifold $(M,g)$ and a smooth function $f: M \to \mathbb R^k$ which is bounded or Lipschitz, you can define a metric $h$ on $N := M \setminus f^{-1}(0)$ conformal to $g$ by choosing $h := (f\cdot f)^{-1}g$. This should make $(N,h)$ complete.