Re: "And the two types of matter might have been expelled unevenly in different directions."
The thing about the Big Bang explosion is that nobody knows the mechanism of how it happened. It's beyond modern physics theories.
The difference between matter and anti-matter is their spin direction. They spin in opposite directions. And that's why they destroy each other, when they come into contact with each other. The amount of friction or energy is doubled, when objects spinning in opposite directions come into contact with each other.
And it's entirely conceivable that the Big Bang explosion itself put a spin on matter and anti-matter. One side of the explosion put a spin in one direction and the other side of the same explosion put a spin in the opposite direction. That's why matter and anti-matter separated without annihilating each other.
They became matter and anti-matter only as a result of the explosion, which also separated them from each other and made them fly in opposite directions.
Our Universe might represent only half of the explosion, and the other half, made of anti-matter, is beyond our ability to see, because it's too far away.