What happens when two creatures actually share the same turn?

Question

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.

I am aware of the rule (thanks to @eternallord66) regarding ties in initiative on page 189 of the PHB which states:

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"?

Answer 1

Short Answer: Since it is your turn where these conditions occur, you as the player that owns that turn, gets to dictate how these affected items work within your turn. Initiative is not needed to be considered, as this is a shared turn, and not tied Initiative of separate actions.

With true polymorph, the bold text does confirm that the polymorphed creature acts on your turn. The sentence right after states:

You decide what action it takes and how it moves

Although the DM resolves those actions, you can also specify how those actions go in relation to your own actions. For example, you can say that you move, and make an attack against the polymorphed creature, then you can direct that it's action is to move, and how you want it to move.

Simulacrum operates very much in the same way.

It obeys your spoken commands, moving and acting in accordance with your wishes

Since you get to decide how it acts, you can make these wishes work out with the rest of your personal actions in any way you'd like. For example, you get to choose if you want your player to attack first, then the simulacrum, or vice versa. Or you can choose to perform all of your player's actions first on your turn, and then the actions for the simulacrum.

The rule for Initiative ties would apply if these are separate turns. These may be actions from two different agents, the player and another creature or effect from a spell. However, the specific rules in these two cases state that all of these actions are now directly controlled by the player, which means that those extra actions are now merged into the Player's turn, at their initiative score. So while the controlled creature is effected, their original initiative score is ignored during the duration of the effect, and their actions are now performed on the player's turn. When the spell effects end, then the controlled creature returns their actions to their original initiative score.

In terms of determining your actions with a shared turn you control, you can decide all actions in any order you choose. Just like you can choose to use your player's move, attack, and bonus actions in any order of your liking. The only difference now, is that you have an additional move, attack, or bonus actions (or maybe all of those) to decide upon before your turn is over.

This is still up for the DM to determine that all of the actions are feasible. For example, if you move the controlled creature out of your player's attack range, and then try to declare an attack against it, the DM may not allow that to happen. So choose which actions to spend in which order-and how you declare them- carefully.

Answer 2

Both spells refer to a defined term.

Turn

From page 189 of the Player's Handbook

A round represents about 6 seconds in the game world. During a round, each participant in a battle takes a turn.

Both are highlighted in bold in the text indicating it is a defined term.

Specific beat General

From page 7 of the Player's Handbook

This book contains rules, especially in parts 2 and 3, that govern how the game plays. That said, many racial traits, class features, spells, magic items, monster abilities, and other game elements break the general rules in some way, creating an exception to how the rest of the game w orks. Remember this: If a specific rule contradicts a general rule, the specific rule wins.

The Spells

True Polymorph

Page 284 Player Handbook

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 DM has the creature’s statistics and resolves all of its actions and movement.

Simulacrum

Page 284 Player Handbook

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.

Implications

• The creature takes it turn at the same time as its controller. Thus any question about when it can take it turn is resolved by deciding when the controlling character can take there turn.
• When the controlling character take their turn both spells explicitly state that the creature takes their at the direction of the controller. Both are worded in a way that is expansive rather than restrictive. In accordance with Specific beats general the controller is free to intermix the creatures action, movement, interactions, and bonus actions with their own action, movement, interactions, and bonus action.

Reactions

page 190 of the Player's Handbook

Certain special abilities, spells, and situations allow you to take a special action called a reaction. A reaction is an instant response to a trigger of some kind, which can occur on your turn or on someone else’s. The opportunity attack, described later in this chapter, is the most common type of reaction.

Both spells speak of only of turns and do not address reactions. However it is also clear from both spells that the creature is now a NPC in its own right.

True Polymorph

If the spell becomes permanent, you no longer control the creature. It might remain friendly to you, depending on how you have treated it.

Simulacrum

The duplicate is a creature, partially real and formed from ice or snow, and it can take actions and otherwise be affected as a normal creature.

Both imply that if it has a reaction it can use it will use it like any other character or creature would.

Roleplaying

As mentioned above both spells give advice on how to roleplay the creatures in question. So outside of combat any thing either of these would do would be adjudicated normally by the referee using the advice found in the spell.

The creature produced by True Polymorph would be friendly if treated well but otherwise normal. The Simulacrum is an automation with limited free will that looks like the target of the spell as well as possessing the target's capabilities, save hit points & equipment, at the time of casting.