My Homeland is Foreign?

[Elmetiacos]

Building somewhere for a governor to hang out in greatly cut the culture penalty, as well as persuading the inhabitants to remodel their homes in the Greek style.

[/Bean\]

does it just take time for the culture penalty to go down, as more of your people move in? or does it drop when you build specific culture buildings such as the festival buildings or theatres if your the greeks?

[Elmetiacos]

The culture penalty dropped from 3 to 1 little black and white masks when the governor's mansion was finished. At the same time, the look of town changed from barbarian to Greek on the map. Of course, I'd had it under my control for long enough to build a homeland government plus the mansion, so what ~14 turns? It might have happened anyway.

[MarcusAureliusAntoninus]

does it just take time for the culture penalty to go down, as more of your people move in? or does it drop when you build specific culture buildings such as the festival buildings or theatres if your the greeks?

It will go down when you destroy or upgrade buildings that were built by a different culture.

[Ludens]

It will go down when you destroy or upgrade buildings that were built by a different culture.

As Elmetiacos implies, the governor's residence is the most important one for culture penalty.

[Conqueror]

Temple buildings are also significant.