US tourists: what is the new strategy for staying more than 90 days?

Good news -- you don't have to leave the country, if you don't want to.  The "new strategy" is the same as the old strategy:  go to the local Polícia Federal office a couple of weeks before your 90 days are up, and request an extension.  As US citizens,  you're still entitled to be in Brazil for 180 out of every 365 days.  Not sure why you think it sounds like a "nightmare" -- most people who have done it describe a pretty straightforward process, and cheaper than flying in and out.

EU citizens,  unless their countries had prior agreements with Brazil that preempt the current EU agreement, are on a 90/180 day plan, which explains why they have to leave  after 90 days.  it still comes out to 180 in 365, but they're required to break it up.

Thanks so much Abthree! Very useful.

The reason I said "nightmare" is that it's my understanding that here in SP you have to go to the PF in Lapa, which I've heard is a memorable experience for its slow lines.

That's why I wondered about the airport, flying back in you might be able to get the extension from the Federal Police upon reentry.

Thank you.

Sure thing! 

I don't know Polícia Federal practices in São Paulo well enough to advise you with certainty, but most of the complaints about the Lapa office on threads here have related to applying for permanent residency, which is a much more complicated process.
Some people have reported success in requesting extensions at airport offices;  might be worth an Uber ride to check out.

oldbroom wrote:

Thanks so much Abthree! Very useful.

The reason I said "nightmare" is that it's my understanding that here in SP you have to go to the PF in Lapa, which I've heard is a memorable experience for its slow lines.

That's why I wondered about the airport, flying back in you might be able to get the extension from the Federal Police upon reentry.

Thank you.

I would not risk waiting as trying to extend on re-entry. May not even be able to board a flight.
As abthree stated go to airport now and ask. They have experienced agents and can help.