If you want to see about becoming a governor (currently, only Yew and Britain have active governors), have adored or better standing in your city of choice, be a citizen of the city, and talk to Topaz after the next meeting. Provided that your account is in good standing, you can typically get made a governor within 2-3 weeks (if not right away). Do note that the character that becomes governor must remain a citizen of the city, and cannot cross-shard while governor (I use one of my crafters to do all my cross-sharding).
Reminder: There were apparently a city or two where people ran for the office, were elected, but never accepted. You don't officially become governor if you win an election until you go to the city stone and accept the position.
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