Based on the Providing services abroad within the EU text, this would depend on if the place of establishment in Country B has been registered there.
The main criteria being:
- how often, for how long and how regularly
If I have understood your scenario correctly, both Country A and B have been registered (Country B being tax resident)
- Country B effectively controlling Country A (which officially is the headquarters)
So the local laws of the country each has been registered in will apply, based on it's activities (organizational forms).
Since the organizational forms can differ radicaly between countries, I don't think an answer can be given without knowing which countries Country A and Country B are.
The Chamber of Commerce and Industry (IHK) link give a general summary of organizational forms in Germany.
It also includes information how to deal with a specific form that does not exist in German law (Representation, Repräsentanz).
General conditions without setup in Country B
If you have a registered business providing services (as an architect or tourist guide, for example) in the country where you live, you can offer those services in another EU country without setting up a company or branch there.
This can be useful if you want to:
- provide the service there only temporarily
- provide the service just to a specific client living there
- test the market before expanding your company there.
In principle, you should be able to supply services in another EU country without having to comply with all of that country's administrative procedures and rules (like obtaining prior authorisation to do business). But you may need to notify the public authorities that you will be offering services in their country. The other country must have valid reasons for imposing its requirements.
General conditions with setup in Country B
Despite this principle, you can't just assume you can provide services without setting up a company locally. Whether you can or not will depend mainly on how often, for how long and how regularly you want to provide services.
Also, different rules might apply to certain sectors, for example:
- financial services
- healthcare services that may only be practiced by members of a - - regulated health profession
- private security services
- gambling services
- notary services
- temporary work agency services
- telecommunications, broadcasting and electronic services
If you choose to register a company in another EU country, you will have to comply with national rules for the incorporation/registration of a subsidiary company, branch or agency and most of that country's rules for setting up a business (including recognition of your professional qualifications and applying for the necessary permits). Compliance with national, European or international standards may also be required.