We already know that virtual matter/antimatter pairs can (momentarily) decouple from each other and exist, where nothing previously was. Perhaps all the Big Split took was for exactly one pair to split. Because when that pair split, they appeared not only as matter and antimatter, but also as spacetime.
At t=zero, there is something very particular and weird about time. During the rest of the life of the universe, time is a line: it starts at t=zero, and extends to now. But at t=zero, time was a point, a zero-dimensional extension. And because probability is always divided by the available time span, the probability at t=zero was divided by a zero. Probability became infinite.
In other words, anything that could happen, HAD to happen. But at that point, before quantum randomness, only ONE thing could happen, which is the spontaneous splitting of other matter/antimatter pairs. And since probability was infinite in that moment, it gave rise to infinite matter and antimatter, in infinite spacetime.