Forum-based.
But the TC should gain importance. It is the most direct way to interact during game-play.
The most likely possibilities that could happen in a TC should be forseen by the elected officials and instructions posted in the Instructions thread in advance. This asks more from our elected officials, but what else are they elected for?
I give you an example; it's the start of a new game; we have 2 cities and some workers. No contacts yet and some units scouting and a settler on it's way to a designated (in the forum) spot. It is likely we'll meet someone soon, or encouter barbarians.Instructions should be posted expecting this. It could be like:
"if you meet anyone, stop the chat",
"if you meet anyone, declare war",
"if you meet anyone, make as many trades as possible",
"if you meet anyone, don't trade Tech XXX, but trade the rest as you like",
"if you meet anyone, prioritise gold over tech; so sell rather than trade techs",
"if you meet anyone, don't try to get tech XXX, we'll research it soon". etc.
Or:
"As soon as you see barbarians, settle on the spot or if they can't kill the settler move to the designated spot to settle",
"As soon as you see barbarians, switch city XXX to unit YYY",
"As soon as you see barbarians, send unit XXX in city YYY to attack / defend unit ZZZ",
"As soon as you see barbarians, stop the chat".
"As soon as you see barbarians, increase lux/sci/tax"
I hope you get the point.
This will make the TC more fluent and prevents (to a certain degree) a DP only playing 1 or 2 turns due to (semi) unexpected events.
A second point related to this is that the TC can continue for several turns, not limiting it to 10. And it makes it clear when to stop a chat and when not to; when a situation occurs that is not expected or posted, that stops it.