Subject to antitrust law is not an inherent property of a corporation. Antitrust law is a statutory construct, as are the tax benefits of incorporation.

What you’re failing to understand is that while there are different types if corporations as legal constructs, with differing advantages and disadvantages mostly related to their overall tax liability, a corporation in the general sense is just a group of people come together for a common goal. Usually that goal is profits. You join up with 5 people putting in 20% of the money, you are entitled to 20% of the future proceeds.

So if you take a distribution of $1000, all other shareholders holding equal shares, they must also take a distribution of $1000, or equal to their share of equity in the company.

A union has a different structure, different legal privileges and burdens, and different goals, but fundamentally it is a group of people with a common goal. I think where they tend to fall down is that unions can grow so large that the power is so diffused amongst individual members that even people in unions may come to resent the union more than their employer, particularly if they are always in the minority on union issues. In which case they might find they are actually more in alignment with their employers rather than their union leaders.

Is that even true? Quick Google searches seem to show several instances of antitrust accusations against non-profits including trade unions and professional associations. There does seem to be a basic exemption for trade unions, but it seems that trade unions can lose that exemption for behaving in certain ways.

Sure, there are exceptions. But the entire purpose of labour unions is to engage in anti-competitive behaviour.

With or without a union, there are no laws I am aware of that prohibit workers cooperating to get higher wages from employers.

That does not violate anti-trust law in the US. It is not a special exemption the union has, it doesn't violate anti-trust law in the US even if there isn't a union involved, for employees to "collude" to get higher wages.

Or if there are such laws I'm not aware of, feel free to share. But I gather you think it ought to be illegal? Why?

Can you elaborate on that? I don't understand what's inherently anti-competitive about labor unions, unless you think that any time humans organize themselves is anti-competitive and the only thing that counts as competitive is a total lack of organization and cooperation above the level of the individual.