So far we've got it decided that the world is just like this, both universes together. Maybe it's always been like that? Or maybe both worlds have large portals on them that connect them together like stargate or something like that. So that way iconic places from both are present but people from each universe enter the world. 

Assuming this could spread to space, those "stargates" are present in random locations around for some reason!

I think in terms of plot we could go a ton of different ways. Depending on how big this is, we have a couple options.

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1. A newly created "police force" for both universes that has to deal with problems in both universes. Joker loose in New York City? Mandarin attacking Star City? We got this! Maybe there's a classic "new injustice league" is created to oppose and maximize chaos. Maybe they're trying to truly connect both universes, an event that would cause reality to just crumble as two universes should never fully coexist. 

2. Galactus, with new sources of food, has set out in the DC Universe to consume new planets. He manages to consume Mogo, the Green Lantern Planet, and gains access to a limited supply of Will. With even more power, he sets out to consume so much more within both universes. Even gaining allies in cosmic villains from both universes. 

3. Mongrul, now with access to more "willing" gladiators for his colesium on his planet, sets out to capture some of the best fighters in both universes to host the ultimate war game. Will our heroes be able to stop him in time? Or will they soon become part of the war games and have to fight to survive or plan the deadly escape?

4. The Brood have spread to both universes and is being led (mind controlled) by Starro to conquer all worlds! With a host of villains and super heroes, as well as the Brood, under his command...will our heroes be able to stop the starred conqueror?! 
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Those are just a few options I can come up with on the spot.