Related to this Q/A ("Does a controlled mount share its rider's turn?") where there doesn't seem to be a consensus on whether or not the rider and their mount actually share a turn.
I am left wondering what even happens when things do share a turn:
The true polymorph spell states:
Object into Creature. You can turn an object into any kind of creature, as long as the creature's size is no larger than the object's size and the creature's challenge rating is 9 or lower. The creature is friendly to you and your companions. It acts on each of your turns. You decide what action it takes and how it moves. The GM has the creature's statistics and resolves all of its actions and movement.
Similarly the simulacrum spell states:
The simulacrum is friendly to you and creatures you designate. It obeys your spoken commands, moving and acting in accordance with your wishes and acting on your turn in combat.
It is notable that these do not simply say that you have the same initiative like the mounted combat rules, instead they say that you have the same turn.
If a tie occurs, the DM decides the order among tied DM-controlled creatures, and the players decide the order among their tied characters. The DM can decide the order if the tie is between a monster and a player character. Optionally, the DM can have the tied characters and monsters each roll a d20 to determine the order, highest roll going first.
However in this case there is no tie in initiative, the creatures are simply acting on each other's turn.
Does this mean that you and the creature/simulacrum can go back-and-forth taking actions and moving throughout this "shared turn"?