The problem was that the linked model did not have any Rooms in the equivalent phase that was set in the host file Views in which the Room Tags were to be placed. Since Rooms are phase-specific, they need to be recreated for each phase.
In this particular case, there were variations in the names that were given to the phases. To verify how the phases for a particular linked file are mapped (and make any necessary adjustments):
- In the host file, select the linked model.
- On the Properties palette, select the Edit Type button.
- In the Type Properties dialog, under the Other category, select the Edit button to the right of the Phase Mapping parameter.
- In the Phases dialog, you can assign the appropriate phase from the linked file to each phase in the host file. Revit will usually match up phases with the same name. (I am not sure what happens if the same names are used, but the phases are in a different order. That is such bad practice that I do not even want to test it.) Selecting in the Phase from linked file column next to a given host Phase will activate a drop-down list, which can be used to select from the phases defined in the linked file.