Deep in an underground lab somewhere in the depths of the earth something breaks and a connection to another world is formed. Something very wrong is about to happen.
Nobody know this, but one percent of the population is going to be swapped from the world we know to a similar alternate universe.
They'll retain their memories, but wake up in the body of themself in that universe. Their alternate universe self will do the same, ending up in our world.
The lab was specifically trying to look at reproduction across the multiverse, meaning that most of the universes people will be swapped to have some sort of genital difference. Something especially unfortunate.