Probably it would be something like as follows:
Have a group of N graders. And a parity of k. Let's say N is 6 and k is 2. Randomly shuffle the assignments and partition the assignments into N groups.
Each grader gets assigned k of the N groups such that they share at most 1 overlap with any other grader and each group is assigned to k people. The assignment orders are shuffled for each grader. They mark up and then grade the assignments.
Then for each of the N groups, randomly shuffle the group and equally distribute the assignments to the N-k graders.
Now each grader reviews the assignment grades/markups (in random order) and assigns a grade based on the k grades/markups from the previous rounds along with a rationale for the grade assigned.
From there the student receives the final assigned grade, the rationale for the grade, and the k markups. If they have a complaint they can go to the professor (who then can also see the k initial grades along with everything else) to dispute the grade for the assignment.
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This way each TA only has to mark up (class size * k / N) assignments, and review (class size / N) assignments to assign a final grade (which should take far less time to do than the initial markups). On top of that every assignment has a guaranteed (k + 1) separate eyes on it. And then the professors can serve as an unbiased arbiter while retaining all the context from the process.
To take it an additional step further, the professors could sample a random subset of the assignments to verify the markup and grading is going properly.
And those reviews/grade adjustments can then be recorded (along with the final grade/rationales) to document how a given TA's grading deviates from the final reviewed grade or the grade the professor assigns. Likewise for a TA's final assigned grade deviating from the professor's. This would allow deviations to be mitigated over time and major deviations to be identified.