It's an article of faith for some on the right that unions wreck productivity.
Not in the private sector. I would be surprised if you had lower productivity in unionized firms. You can't have both lower productivity and higher wages in unionized firms--they would go out of business.
In the private sector, a union should raise wages. When it raises wages, the firm should substitute capital for labor, raising productivity.
The way to think about dysfunctional work rules is as a tax on labor--causing the firm to substitute even more capital. Other things equal, the dysfunctional work rules reduce productivity, but other things are not equal.
In the private sector, I would look for the adverse economic consequences of unions to show up in the inter-firm allocation of labor. Firms in the union sector will pay too much in wages and hire too little labor. The workers they don't hire will end up either in the non-union sector (where they will be less productive) or unemployed (where they will be really less productive).
In the public sector, I can believe that unions lower productivity. But I would be surprised if it happened in the private sector.