The final fallback for market resolution (after all reporters have reported on an event and someone still believes the answer is wrong) is called Fork Resolution. Only one fork can be active at a time per branch (in Augur, a branch is akin to an entire universe that is isolated from other universes). Once a fork starts, all other markets that are awaiting reporter based resolution (this does not include automated reporting) are delayed until the fork resolves. Once the fork resolves, all of those pending markets will migrate to the branch (universe) that won the fork.
The fact that existing markets migrate to a particular world view is one of the first differences with AgnosticPM. In AgnosticPM, the universe a market lives in is based entirely on markets that came before. In Augur, while a market is created within a particular universe (branch), it will migrate forward to another branch if a fork occurs before the market settles. The reason for this is because if a fork occurs, we must assume that reporters have been compromised and any reporting they do is invalid. It is also assumed that the fork resolution process is the best answer we have to "who was right".
As for how the fork resolution process actually works, its pretty simple. Before the fork, there is REP (tokens) that is associated with a particular branch (universe). This REP is traded on exchanges and has some market value. When a fork occurs, a new branch is created (conceptually) for every possible outcome and each branch has a new REP token. Anyone who holds the original REP token can convert (1:1 ratio) their token to a REP token on any branch of the fork. The act of converting your REP from the parent branch to a child is how you vote for which branch you think represents reality. Once you have cast your vote, there is no going back so be certain!
At the end of the fork (currently ~2 months) whichever branch had the most REP migrated to it is considered the winner and all existing markets needing reporting will migrate to that branch, and share tokens for the forking market will be redeemed using the outcome from that branch.
This begs the question, why would I risk my REP casting a vote instead of just keeping my original REP? The answer to this is that no new markets can be created on the original branch and all existing markets will resolve on the winning branch. This means that REP on the original branch will yield no reporting fees to REP holders as there will be no markets to report on. As a REP holder, your job is to pick the branch that you think the most future fees will come from. This means you want to pick the branch that has the most market participation (volume/outstanding interest) so your REP gets the maximum amount of fees long term. In almost all cases, this will be the branch that represents the truth/reality. The only cases where this is less certain are forks that are highly contentious, where not everyone agrees on what reality even means.
In order to incentivize people to migrate their REP during the voting period (tentatively 2 months), there will likely be some small inflation assigned to everyone who migrates sooner rather than later (e.g., 5%). The idea here is that if you pick a side (rather than abstaining) then your share of REP will not be diluted. If you abstain, then your share of REP will be diluted by some amount relative to how many people migrate. The goal here is to make it so abstaining from the vote is a losing strategy in almost all cases (non-contentious forks) as this increases the cost of mounting a forking attack against Augur.
Regarding AgnosticPM, Augur is similar in that it has a concept of branches (universes) which have a history of truths associated with them. The primary difference between Augur and AgnosticPM is that Augur has a concept of finality, meaning that the system has the ability to assert that a particular outcome is an absolute truth for the sake of paying out winnings. In AgnosticPM, there is no way to extract deposited ETH from the system, its ETH-in-only. This is a pretty huge difference as it means that AgnosticPM, while fun and possibly effective, doesn't have a way of liquidating its ETH holdings back to participants to payout. As long as AgnosticPM experiences heavy growth, this may not be a problem but if it ever ceases to experience usage growth then this becomes a big problem. Investors know this, which will likely make them hesitant to buy any AgnosticPM share tokens, which is likely to drive their value to below 1 ETH (their target).