What is the role of 'for them' in the following sentence? Could it be omitted without any change in meaning?
The only way for unions to maintain their clout would be for them to organise internationally.
(emphasis by me)
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In this case, it could be missed out. That's because the unions are the effective subject of maintain their clout, and thus they are also the implied subject of to organise internationally. They are the only possible subject in that sentence.
However, consider the sentence:
The only way we'll see unions maintain their clout would be for them to organise internationally.
In that case, if it weren't for for them, it could be that the sentence is suggesting that we organise internationally, rather than the unions. The relative pronouns them indicates that it's the unions, and as they are the only third-person in the sentence they are the only possible antecedent of them.