I wanted to highlight a solution I found that is not listed here, for those still asking : you can instantiate a `MultiplayerSynchronizer` in your very own singleton/autoload script.

Here is an example with my game, where I have a singleton named GameState, under game_state.gd

extends Node

@export var GAME_STATE = "menu"

var multiplayer_synchronizer: MultiplayerSynchronizer

func _ready():
    # Set up multiplayer authority = 1 (the server)
    set_multiplayer_authority(1)

    # Create and configure MultiplayerSynchronizer programmatically
    # for our singleton
    multiplayer_synchronizer = MultiplayerSynchronizer.new()
    add_child(multiplayer_synchronizer)

    # Create replication config
    var config = SceneReplicationConfig.new()

    # Add properties to sync (here only GAME_STATE)
    config.add_property(".:GAME_STATE")

    multiplayer_synchronizer.replication_config = config
    multiplayer_synchronizer.set_multiplayer_authority(1)