How do you play New Angeles Sol's ability triggering re-playing Predictive Algorithm in a multi-access situation with multiple agendas. (for instance multiple agendas in Archives)

I think that the each agenda stolen triggers the New Angeles Sol ability which allows the corp to play Predictive Algorithm again before the next agenda is accessed.

I haven't been able to find anything that contradicts this, but the Official FAQ doesn't say anything about New Angeles Sol or the timing of triggers

You are correct. It comes under Chain Reaction rules in the FAQ. Each time an agenda is stolen it triggers NASol. Each trigger must be resolved before continuing to the next access.