I briefly worked as a field service engineer before becoming a programmer. The job that I worked was non-union but many (all?) of the electricians where in the local union.

Do you know how that works with the idea that by joining a union you are giving up your right to work non-union jobs?

I'm not trying to say that you are wrong, I just don't know enough to know if that was a special case or if there are certain circumstances where a union worker can or cannot work a non-union job.