[notforever]

No good reason for test errors to be uncorrelated, ie. quite possible same people who trigger false positives on the first test would be false positives on the second etc.

[subroutine]

Sure, there will probably be some correlation. Depends heavily on how the 3x testing is designed.

T1. Drive up saliva swab by nurse, for serology test.

T2. Re-deploy census peeps to get samples (and census info) from entire neighborhoods.

T3. 23andMe-like home kit express mail to lab

The odds of people getting a FP on all three might not be 1 in 1000, but it probably isn't 1 in 10.

Anyway, as it is, leading 1 in 10 people to falsely believe they are immune, isn't much better than having no test at all.