So this might not occur depending on the path you take but in case you've not encountered this yet it's good to be forewarned, unfortunately I do not have a photo to show you this time so I'll have to explain it as best I can.
If you have done every optional part of the quest so far, after you return from the food handouts and speak with the King, one of his henchmen will stop you from telling him about the envoy. For me, I managed to get caps out of him to keep my silence, but then decided to tell the King anyway.
This is where the problems start...
If the King is sat in the theatre, Rex will be at his right hand side, very close to the King and the table.
If you tell King everything, including the envoy that was sent he'll say something like.
"Whoa now... tell them that I'm willing to co-operate... WHAT THE HELL!?"
At this point, the Hud will vanish and another henchman will magically appear between Rex and the table. The henchman will try and run past Rex... but he can't.... The player is now stuck in an void. The path that the NPC tries to take is to go through Rex and around the back of the King to the front of the table, only when he gets here does the next set of dialogue choices begin. However.... he can't. Why not? Because NPC's cannot force move other NPCs out of the way... unlike the player.
Meaning that this guy will continue to run into Rex and the player will be stuck until that NPC gets past REX and around to the front of the kings table the player cannot continue on with any part of the game, reason being is that he's locked in a conversation and cannot end it until this new NPC begins his dialogue with the King.
There is however... some good news.
If this occurs, you can reload the auto save and you'll start inside the Kings HQ just before you enter the theatre.
To avoid the NPC getting stuck, you need to try and navigate Rex away from the table... which is not an easy task because his AI restricts him to being close to the King, and any attempts to 'push' him out of the way will be countered by the AI.
One way I've found to move him is to stand away from him, and try to talk with him... he tends to move that way (but not much).
Or.... get rex as your companion so he won't be there when this NPC magically appears out of nowhere.
This bug is very annoying due to the fact that the dog is not easy to move out of the way, unlike other NPC Neutral dogs, this one doesn't like being moved around, meaning you might have to use the auto save a lot before you get the dog to move out of the way.
So if it happens to you, just keep calm and try to move that darn dog.
This happened to me at around midday-afternoon time period, so if you try it at night the King might be in another spot or the NPC may come from another area and not get stuck.
