Assuming the plan's trust is a domestic trust, can a foreign corporation sponsor a 401(k)?

What do you mean by "foreign corporation" ?

For example let us say Bayern AG, a German corporation, cannot just start business in the US with US employees, as it feels. It has to register in a state etc and get an IRS EIN number etc etc. No one is able to just start doing business when, where or how they feel, foreign or US.

Assume the foreign corporation has a branch office in the US. They have an EIN, and operate under a fictitious name, but they do not register under state law. The only business the US branch office conducts is sales for the benefit of the foreign corporation.

Assume further that the US branch office is not considered an "employer" under 401(a) despite the EIN and issuance of W-2s to US branch salesmen.

Why that assumption? Isn't that really your question?

A non-U.S. employer but that derives income generated in the U.S. from employees in the U.S. may have a 401(a) QRP. There are Revenue Procedures that once detailed a different address for determination letter applications if the QRP was sponsored by a foreign employer, and past IRS Form 5500 instructions mention a foreign employer in certain regards.

Registering as a foreign corporation in one or more states in the U.S. is a matter of state law (which the foreign company may be in violation), but it has an EIN and has a U.S. payroll for those U.S. branch salesmen. It is an employer for 401(a) purposes.