Here is the opinion in  Adams v. Comm’n on Appellate Court Appointments.

An excerpt describing the tribal judge in question, Paul Bender:

Bender, an independent, stated on his application that he serves as “Chief Judge of two Arizona tribal courts.” Bender, a law professor at Arizona State University, serves as the Chief Justice of the Supreme Court of the Fort McDowell Yavapai Nation and the Chief Judge of the Court of Appeals of the San Carlos Apache Tribe.

And from the analysis:

Consistent with the tribes’ distinctive status, Arizona’s constitution and laws generally do not include tribes within the meaning of the word “public.” Cf. Ariz. Const. art. 20, § 4 (referring separately to “public lands” and “lands . . . owned or held by . . . Indian tribes”). As noted above, see supra ¶¶ 23-29, Arizona’s constitution and statutes refer in many places to public office or public officers (for example, in provisions governing recall or financial disclosure), but none of those provisions has been construed to embrace tribal offices. Indeed, at oral argument, counsel could not identify any instance in Arizona law in which the word “public” has been interpreted to refer to Indian tribes.

From the dissent:

Giving the term “public office” the broad construction that § 1(3) suggests, I would conclude that Bender, as chief justice of two tribal courts, holds public office. At oral argument, amicus Valley Citizens’ League’s counsel (advocating for Professor Bender’s eligibility) expressly stated that Bender is a public officer of the respective tribes he serves. The constitutions and bylaws of both the San Carlos Apache and Fort McDowell Yavapai tribes support this acknowledgement, expressly delegating the judicial authority of their respective nations to their judiciaries. And it is indisputable that the judicial powers of a tribal nation are governmental powers of a sovereign. See 25 U.S.C. § 3631 (2006) (recognizing inherent sovereign authority of each tribal government’s judiciary); Penn v. United States, 335 F.3d 786, 789 (8th Cir 2003) (“[A] tribal court judge is entitled to the same absolute judicial immunity that shields state and federal court judges.”). As a judge, therefore, Bender exercises a portion of the governing power of these two sovereigns, making him a public official of these tribes.