Also note that the information presented to the controller can vary significantly depending on the radar system installed.
(Controllers have to do more or less the equivalent of a type rating to 'master' the system they are working with, which is usually common within a state, but across different countries, there are many different systems.)
Some research into the system in use at the particular location you're setting the story in would be indicated.
As an example, there is a thread
here concerning a middair in Brasil. (It's big. Page 31 on gives some insite.) (Some more info
here )
The information presented to the controllers on their system was such that it was not immediately apparent to them that one of the aircraft had an inoperative transponder.