I have conflicting branches, feature_x branched from main.

Let's say when rebasing feature_x on current main, while resolving conflicts, I decide to take some (not all) of "their" (i.e. main) files as-is. How do I do that?

I tried:

git checkout main:foo/bar.java
fatal: reference is not a tree: TS-modules-tmp:foo/bar.java
git checkout refs/heads/main:foo/bar.java
fatal: reference is not a tree: refs/heads/TS-modules-tmp:foo/bar.java

You want to use:

git checkout --ours foo/bar.java
git add foo/bar.java

If you rebase a branch feature_x against main (i.e. running git rebase main while on branch feature_x), during rebasing ours refers to main and theirs to feature_x.

As pointed out in the git-rebase docs:

Note that a rebase merge works by replaying each commit from the working branch on top of the branch. Because of this, when a merge conflict happens, the side reported as ours is the so-far rebased series, starting with <upstream>, and theirs is the working branch. In other words, the sides are swapped.

For further details read this thread.