Andrew Kerr and Francis Teal at CSAE have an interesting paper exploring the differences between public and private employees in South Africa.
Unionised public sector and formal private sector workers earn more than informal sector workers - the question is whether this is just because they are simply "better quality" or more productive workers and earn that extra pay, or whether the labour market is "segmented" and trade unions keep wages artificially high and erect barriers to competition from all those informal sector workers.
To explore these competing hypotheses they control for a bunch of individual characteristics which might indicate the "quality" of the worker to see if an unexplained residual remains which we can attribute to labour market segmentation. This includes controlling for "unobserved" but fixed individual characteristics, which is a pretty cool technique you can use when you have a dataset tracking the same individuals over time.
Their analysis shows that the higher wages for private sector unionised workers can be entirely explained through individual characteristics. They are just higher quality, more productive workers.
The higher wages for public sector unionised workers can't be explained this way. Similar workers seem to earn more in the public sector than they would in the private sector.