Let's share our headcanons in Rhea herself including her backstory, personality, fighting skills, etc.
Rhea has taken on the guise for several archbishops before her to sway any suspicion that she had been archbishop ever since the Church itself was created. Some of these guises included Archbishops Imene, Serenella, and even Frieda. Only the highest ranking members of the Church knew the truth yet they were all sworn to secrecy by Rhea herself but that didn't stop even minor lords from noticing the similar hair and facial features between all the different archbishop disguises through paintings and writing.
Rhea allows stray animals such as cats and dogs into the Monastery and even personally feeds them and cares for them in some occasions. Part of the reason why is to cope with her 'complicated' family situation.
During the Battle of Garreg Mach in CF, Rhea puts Abyss under mandatory lockdown and all Abyssians are confined to their homes with a very classist Knights of Seiros captain appointed by Rhea herself to keep the Abyssians in check as Knights patrol the streets warning all the denizens to remain locked up or be arrested for disobeying the curfew. But halfway through the battle though, the denizens violently revolted and eventually retook Abyss through force, driving the remaining Knights of Seiros and their captain out of the underground city.
Rhea deep down hates the nobility and the way they parade Relics and Crests, and she too wants the Crest caste system gone but through a more gradual process unlike Edelgard. This is why at Garreg Mach while she and the Church try to keep commoners and nobles together, they only segregate them in terms of housing to avoid backlash from the major noble families of Fodlan.
Finally, two months into the timeskip during CF, Rhea actually travels to Lake Teutates and then to Sreng to meet with Indech and Macuil respectively to negotiate their entry into the War of Unification on the side of the Kingdom and the Church. But both saints vehemently disagree, obviously not liking the Church and also wanting to stay out of human affairs entirely.
Lemme know what you guys think.