Travellers on the spaceship don’t actually see (or in any other way directly experience) the sudden ageing of those on earth when they turn around, but rather they just infer it from what they do see. Just before the turn-around, what the travellers see of Earth is actually not what they think is happening in their version of “now” but rather what was occurring at an earlier time when the light signal was emitted (and what for them is actually occurring “now” hasn’t been seen yet). After a quick turn-around what they see of the Earth is essentially the same as just before, but now the correction for light travel time is different and the result is that it now seems to them that the events occurring “now” on Earth (which they haven’t seen yet) correspond to a later Earth time than they did just before the turn. So even though they don’t see the jump it seems to the travellers as if the Earth’s clocks quickly jumped ahead. This solves the mystery of the age difference because the amount of the apparent jump in Earth’s time is exactly twice the amount of progress lost during the two periods where the Earth seems to be ageing slowly so at the end of the trip the tavellers and homies all agree that the Earth has aged more.