Mae'r cyfrifoldeb dros farnwriaeth Tribiwnlysoedd Cymru'n cael ei arfer drwy gyfuniad o swyddogaethau Gweinidogion Cymru a swyddogaethau'r Arglwydd Ganghellor. Ers 2015, lle bo gan Weinidogion Cymru gyfrifoldeb swyddogol dros benodiadau, mae'r broses o recriwtio aelodau tribiwnlys - yn aelodau cyfreithiol neu aelodau lleyg - wedi'i chynnal gan y Comisiwn Penodiadau Barnwrol, sy'n gorff annibynnol. Er bod yr aelodau'n cael eu penodi'n ffurfiol gan Weinidogion Cymru, maent yn cael eu hargymell gan y Comisiwn. Mae hyn yn adlewyrchu'r trefniadau sy'n bodoli i'r Comisiwn gynnal proses recriwtio barnwriaeth llysoedd a thribiwnlysoedd ar ran yr Arglwydd Ganghellor. Bydd yr aelodau'n gwybod bod proses recriwtio benodol yn ei lle ar gyfer Tribiwnlys y Gymraeg, lle mae'n ofynnol drwy reoliadau i Weinidogion Cymru ystyried yr angen i gynnal egwyddorion annibyniaeth y Tribiwnlys a rheolaeth y gyfraith wrth benodi'r aelodau.
Responsibility for the judiciary of the Welsh Tribunals is exercised through a combination of the functions of the Welsh Ministers and the Lord Chancellor. Since 2015, where the Welsh Ministers have the formal responsibility for appointments, the recruitment of tribunal members, who may be legal or lay members, has been conducted by the independent Judicial Appointments Commission (JAC). Whilst the formal appointment of members is made by the Welsh Ministers, they are made on the recommendation of JAC. This mirrors arrangements for the JAC to conduct recruitment of court and tribunal judiciary on behalf of the Lord Chancellor.  Members will be aware that a bespoke recruitment process is in place for the Welsh Language Tribunal where the Welsh Ministers are required by the regulations to have regard to the need to uphold the principles of independence of the Tribunal and the rule of law in the appointment of its members.