How to present a tree in which objects may appear more than once?

I have a window showing a number of objects that can be individually hidden or recolored. Accordingly, in a sidebar each has a checkbox and a color swatch that launches a color picker when clicked:

example tree view

However, as shown, the objects also occupy a structure that resembles a (Unix) filesystem: the user has organized them into groups ("directories", but not nested), and each may have more than one name in the same or separate groups. No name or group is privileged or canonical. (The number of groups is almost always 5 or fewer, likely with no more than 25 entries each. Any one object probably has no more than 6 names in total.)

Thus the groups and the names of the objects form a tree that is important to the user, but adding the objects themselves (as "children" of their names) weakens the structure to that of a DAG. The rendering is of the objects, so it is meaningless for a single object's various names to have different colors or visibilities.

What controls can I present to the user that

  1. convey the hierarchy of groups and names,
  2. convey the connection among different names for the same object,
  3. display the colors in association with every name (to allow quick consultation without any gestures), and
  4. minimize confusion from synchronizing the properties "of the names"?

The example image shows one simple strategy: a in group bar and a and b in group foo are all the same object, while two other objects have just one name each. The layout certainly satisfies #1 and #3. However, #4 isn't good: clicking any of the three checkboxes for the shared object must update all three (and similarly for the colors). Moreover, this sameness is only reliably detectable by interacting with them (so #2 is not met); it might be mere coincidence that they have the same color and checkbox state, and the same name ("a", here) might or might not refer to different objects in different groups.

Note that no editing of the structure or names ever takes place; this interface controls drawing the objects and is not involved in creating, deleting, grouping, or naming them.

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