I think this article puts it decently:

5) Is there such a thing as an illegal strike?

Absolutely, though a more accurate term is “unprotected.”

Basically, it's not necessarily that workers would be criminally charged for participating in such strikes per se, but that the law doesn't protect them from being retaliated against in other ways. For example, sit-in strikes don't supersede property owners' rights to have people removed for trespassing.

That said, IMO the Taft-Hartley Act's prohibition on secondary strikes should absolutely be considered unconstitutional under the 1^st^ Amendment right to freedom of association.