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There doesn't seem to be anything on the web about how they determine common opponents among 3 teams.
I'm guessing that they look at the records of common opponents. The Patriots and Phins would be tied in the 10-6 scenario we are discussing. The Jets would have the worst common opponent record.
They would drop off.
I don't understand the confusion... probabaly noone saw a need to put anything explicit on the web because there's really only one logical way it can work. In resolving a three-way tie, they only look at teams all three have played against, and compare each team's percentage against those common opponents. There's no requirement that the number of games be the same, it's the winning % that matters.
Then it would be Phins versus Patriots for common opponents, and the Patriots would win that tiebreaker now that the Jets are out of the equation because the Phins-Patriots have more common opponents than the Jets-Phins-Patriots.
Technically, once the Jets drop out and it becomes a two-way tie, they go back to the first tiebreaker, and head-to-head might decide it. (Not in this particular case MIA-NE, but in general, I am saying.) Where I think you are wrong is the common opponents will be the same (have to be, since MIA and NE are in the same division) except now the Jets are added in. And adding the Jets back in makes no difference, since if one team had an advantage against the Jets it would have come out earlier in the division record tiebreaking step.