Earlier quoted context omitted.
> The anti-union FUD campaign in the US has contributed significantly to the erosion of middle class wages It's way more complicated than that, and it can't simply be dismissed as a "FUD campaign". Union members are actually disproportionately likely to support things like right-to-work laws, which unions dislike because it (more or less) means workers aren't obligated to be members if they choose not to. Similarly,…
Do multiple competing unions happen in the same workplace happen in the US ? As far as I have seen, that seems fairly common in the countries where I have lived (Europe only). And are there some cross-category unions (where workers in different areas and jobs are members of the same overarching organization, which sometimes organizes nation-wide demonstrations or even strikes)?
No, and that's basically by law. Unions lobbied (successfully) for laws which grant them permanent, exclusive access to the bargaining unit once 50% + 1 vote in favor of it.
Pretty much the only cases where you'd have separate unions are where they're representing different types of workers altogether (ie, nurses vs janitors) or when they're not actually employed by the same organization (ie, two different sets of subcontractors).
> And are there some cross-category unions (where workers in different areas and jobs are members of the same overarching organization
Yes, in fact almost every union is affiliated with a federation like the AFL, UAW, Teamsters, etc.