Initially I wanted to call this question "Categorification of meromorphic functions?" but discovered so many questions about categorification that I became scared and decided to replace it with a more obscure title.

The question resulted from contemplating my own answer to Modern treatment of Dirac monopoles and related topics.

Let us work with some version of algebraic varieties. Let $X$ be one such.

Given, say, a regular function on $X\setminus\{\text{a point}\}$, this function might have a pole of some order at that point, or an essential singularity. This is absolutely classical, one way to deal with it is to understand regular functions as morphisms to $\mathbb A^1$ and meromorphic functions as morphisms to $\mathbb P^1$.

Suppose given a line bundle on $X\setminus\{\text{a point}\}$. What can happen at that point?

The analogy with functions might go like this. View "regular" line bundles as morphisms to $\mathbb P^N$ for large enough $N$, or maybe to $\mathbb P^\infty$ which is defined as an ind-scheme obtained from the union of all the $\mathbb P^N$ in a certain way.

Is there some $\mathbb E^\infty$ that is to $\mathbb P^\infty$ as $\mathbb P^1$ is to $\mathbb A^1$? That is, such that $\mathbb P^\infty$ maps to $\mathbb E^\infty$, representing a hypothetical map (functor?) from "regular" line bundles to "meromorphic" line bundles?