Article 40(2) of Directive 2014/59/EU (BRRD) states that a bridge institution must meet two requirements.
The first condition is that “it is wholly or partially owned by one or more public authorities”. It follows that at least some ownership of the bridge institution needs to be retained by a public authority (including but not limited to the resolution authority or the resolution financing arrangement).
The second condition stipulates that the resolution authority must control the bridge institution but it does not prescribe how.
Against this background, and given that a minimisation of the use of public fundsis a key component of the resolution regime, a solution that allows control of the bridge institution with minimum ownership (for example by means of a “golden share”) is acceptable under the provision of Article 40(2) BRRD. There are however other mechanisms available to comply with the two requirements.
The same logic applies for the asset separation tool (Article 42(2) of Directive 2014/59/EU (BRRD)).

Disclaimer:

This question goes beyond matters of consistent and effective application of the regulatory framework. A Directorate General of the Commission (Directorate General Financial Stability, Financial Services and Capital Markets Union) has prepared the answer, albeit that only the Court of Justice of the European Union can provide definitive interpretations of EU legislation. This is an unofficial opinion of that Directorate General, which the European Banking Authority publishes on its behalf. The answers are not binding on the European Commission as an institution. You should be aware that the European Commission could adopt a position different from the one expressed in such Q&As, for instance in infringement proceedings or after a detailed examination of a specific case or on the basis of any new legal or factual elements that may have been brought to its attention.