In England there is a system of tribunals as well as various courts. Are they entirely adversarial as well as the courts are?
Neither courts nor tribunals are entirely adversarial.
The functioning of a tribunal lies on a spectrum from adversarial to inquisitory. For an example of the latter, see the Refugee Protection Division discussed in Thamotharem v. Canada (Minister of Citizenship and Immigration), 2007 FCA 198:
Parliament had chosen an inquisitorial procedural model for the determination of refugee claims by the [Refugee Protection Board], in the sense that there is no party opposing the claim ... the task of probing the legitimacy of claims inevitably falls to the [officer], who questions the claimant on behalf of the member, and/or to the member of the [Board] conducting the hearing, especially when no [Officer] is present
Even courts have inquisitorial aspects. In a bail hearing (Criminal Code, s. 518(1)(a)):
the justice may, subject to paragraph (b), make such inquiries, on oath or otherwise, of and concerning the accused as he considers desirable
A tribunal is not a court and does not necessarily operate the way a court does. It operates the way Parliament has set down for it in the enabling legislation and regulations.
Usually, a tribunal is either given broad powers to establish the “facts”, allowing it to be more inquisitorial than a court can be, or it may be tightly constrained, forcing it to be even more limited than a judge.