Their Phase Intent was to change the dominant government. They played well (well actually my dice are just laughably bad) and managed to remove the church and place their family as the head of state.
Next session they are about to lose the phase by a wide margin.
How am I supposed to resolve this in the fiction, it does not seem to fit?
Play it by the rules. There’s still some reason why the church holds power - the family is nominally in charge but they don’t actually have any power (the British crown), or they do have power but the church is pulling the strings in the back (the US government and the corporate lobby). Give them what they want, then when you win the phase tell them in the epilogue how it really went down.
It’s practically Dune-like! They think they hold power. They think they’ve won, but just as they settle in either they’re swept from power by a coup or there’s a popular uprising or a battalion of Sodalis drops out of distortion and an Inquisitor-Legate sets up shop in a new capital and declares them apostate. Save all of this delicious color for the wrap up at the end of the phase. I bet they’ll want to play another one.