I want to build a game with a concept similar to Ingress, where the game map is overlaid on a real one.

Where can I find map data for an app like that? How (generally) do I integrate my own game objects into that map? Game objects need to move in real time, so having them as static markers on the map isn't sufficient.

I thought I could use Google Maps but as far as I can tell it would be impossible to create my own graphical style (like Ingress has done) with the official API provided by Google.