Let me start off by saying I know this idea might have been requested before, but I would like to add my interpretation.
What is a Clan War Tournament?
A clan war tournament would consist of 8/16/32 clans. The winning clan advances. Besides that, it is a normal clan war.
How would it work?
In war search screen, clans can choose between Start War and Join Tournament. If they choose tournament, Supercell will start trying to find equally matched clans. Once 7 (or more depending on how many clans Supercell decides to put per tournament) other equally matched clans are found, they are randomly put into a bracket. The wars start at the same time. Same prep day, same battle day. The winners advance and immediately go into the next war.
Wouldn't it take a long time to find a match?
Yes, but not as long as you might think. If there are 8 clans in the tournament, then imagine you are one of those clans. If you were to find 3 wars in a row, you must find 3 other clans. If you were trying to find a tournament (3 wars in a row, 1 search total) you must find 7 other clans. Theoretically, this should take about twice as long.
What are the advantages of a tournament?
There are 2 main advantages: rewards and glory. First advantages. It would work something like this. Round 1, normal loot/xp bonus. Round 2, +25% loot/xp. Round 3, +50%. Etc...
As for glory, think about it like this. For a 16 clan tournament you participate in 4 wars. In normal circumstances, you could say you were better than 3 other clans. In tournaments, you would be able to say you were better than 15 other clans! This would be appealing to top war clans trying to prove their dominance, and also new war clans, trying to show their skills.
What happens when you lose?
Simple. You can either go start another tournament, start a normal war, or have a few farming days.
How many clans would there be per tournament?
That is totally up to Supercell, but I would suggest 8, 16, or 32. The more clans, the longer the search, but the more loot bonus you would need to add. Also bigger tournaments would give you a better chance to face top war clans.
If you have read this far, thank you. Please post any questions or concerns in comments and I will add them to the original post. Thank you! :)

