The Defining Tension blog posts on "the right of inquiry," which we learn "is a very popular and important avenue for corporate litigation in the Netherlands; the framework as such does not exist in other countries." Certainly, I'm not aware of any US corporate law equivalent. The post by Bastiaan Assink relates that:
The right of inquiry entitles shareholders - and trade unions, as well as some others, including holders of depository receipts - who own either 10% of the outstanding stock or shares with a nominal value of at least Euro 225.000 (or less if provided by the corporation's articles of association), to request the Enterprise Chamber of the Amsterdam Court of Appeal to review the course of action followed by/the affairs of certain legal entities, including corporations. ...
In the so-called 'first phase' of inquiry proceedings, one of the central questions is whether there are 'well founded reasons to doubt the correctness of the course of action followed by the corporation' (section 2:350(1) DCC). If the answer is yes, the court may order an inquiry - after having balanced all the interests involved, including that of the corporation - and appoint one or more investigators. Other important questions are (a) whether the plaintiff is authorized to make the request (typically: does he meet the shareholding threshold?) and (b) whether the plaintiff (i) has informed the management board and supervisory board in writing of his complaints first and (ii) has given the corporation a reasonable period to investigate the complaints and take the necessary measures. The first phase ends when the investigators file their report with the court. This first phase is an independent proceeding: if the plaintiff wants to proceed with the inquiry proceeding, he will have to file a new request, which starts the so-called 'second phase'.
In the second phase of inquiry proceedings, the court determines upon request, based on the investigators' report, whether the corporation has followed an 'obvious incorrect course of action' (section 2:355 DCC). This can be framed as clear mismanagement. Upon request, and if such mismanagement is established, the court can take one or more measures to put an end to this mismanagement. The court can for example nullify decisions taken by organs of the corporation, fire directors, or even order the dissolution of the corporation. The list of possible measures is limited....
... the right of inquiry does not center around liability related questions, as it is not a proceeding primarily designed to establish liability of actors in the corporation for damage suffered by the corporation. If litigants want to initiate liability proceedings, they will have to address a different court in different proceedings. Sometimes inquiry proceedings are used as a stepping stone for liability proceedings, as it may be easier for a plaintiff to make his case in liability proceedings with the investigators' report in hand. Although a determination of mismanagement in inquiry proceedings does not automatically result in director liability, such a determination will generally give a plaintiff in liability proceedings a head-start, as the conduct reviewed by the Enterprise Chamber will typically be more or less the same as the conduct complained of in liability proceedings and the Enterprise Chamber is generally seen as a court specialized in business matters.
I suppose the closest analogy in US law is the provision in Model Business Corporation Act section 8.09(a) for judicial removal of directors, but that provision requires much more serious misconduct than mere mismanagement:
The ... court of the county where a corporation’s principal office (or, if none in this state, its registered office) is located may remove a director of the corporation from office in a proceeding commenced by or in the right of the corporation if the court finds that (1) the director engaged in fraudulent conduct with respect to the corporation or its shareholders, grossly abused the position of director, or intentionally inflicted harm on the corporation; and (2) considering the director’s course of conduct and the inadequacy of other available remedies, removal would be in the best interest of the corporation.
The commentary to that section further indicates that policy disputes and business judgment arguments should be left to the shareholders usual corporate governance powers such as voting and suing for damages. In addition, this provision hardly could be described as "popular" or "important." To the contrary, judicial removal of directors is rare. After all, why should we think that judges (or shareholders, for that matter) are well-positioned to evaluate the merits of board actions? To the contrary, the logic of the business judgment rule is that, as I argued in The Business Judgment Rule as Abstention Doctrine, Justice Jackson famously observed of the Supreme Court: “We are not final because we are infallible, but we are infallible only because we are final.” Neither courts nor boards are infallible, but someone must be final. Otherwise we end up with a never ending process of appellate review. The question then is simply who is better suited to be vested with the mantle of infallibility that comes by virtue of being final—directors or judges?
Corporate directors operate within a pervasive web of accountability mechanisms. A very important set of constraints are provided by a competition in a number of markets. The capital and product markets, the internal and external employment markets, and the market for corporate control all constrain shirking by directors and managers. Granted, only the most naïve would assume that these markets perfectly constrain director decisionmaking. It would be equally naïve, however, to ignore the lack of comparable market constraints on judicial decisionmaking. Market forces work an imperfect Darwinian selection on corporate decisionmakers, but no such forces constrain erring judges. As such, rational shareholders will prefer the risk of director error to that of judicial error. Hence, shareholders will want judges to abstain from reviewing board decisions.