The best example that I was given of God being Omnipresent is:
Build a universe in your own mind. Build every detail to it.
What's the relationship between you and that universe? It all exists by your will. The moment you take away your will for a person to exist they cease to exist. Everyone is aware of your presence, there isn't a place/time that you're unaware of. Nor anything that anyone does in this universe affects you (the God). Like if a nuclear bomb goes off or if a fly goes to sleep or if the sun explodes, it's all the same to you as its creator.
God and its creation, both exist, but the presence of God is independent of any other being. While the presence of the creation is all dependent on God. Not that God is in a specific realm. He isn't, but just that he isn't defined/limited by place. You and I as humans are limited/scoped.
This wasn't a perfect example, but to some extent it shows what it means by saying God is present everywhere but not physically...