How to stay in Brazil for 5 months with a (tourist) visa?

In the past, it was fairly common to do "border runs" - when your 90 days were up as a backpacker, you'd exit the country for a few hours, and come right back in. I met many people doing this in most South American countries.

In the past, the common way would be to do a border run. Head over from Foz Iguazu to Iguazu, spend the day checking out the falls, maybe spend a night, and then re-enter. That's 90 + say, 89 days (one in Argentina), and 179 days is still more than 5 months :)

However, as you've mentioned, the tourist visa has a rule now that you can only be in Brazil for 90 days out of 180. So it's quite clear, if you want to be a legal visitor, without overstaying, the only legal way you can manage this is to get a non-tourist visa.

Bear in mind, as well, if you're on a tourist visa and interning, and they don't allow that on a tourist visa, it's definitely in your best interests to get a work or student visa. Just because it's time-consuming is NOT a valid excuse for breaking the law, and if officials work out what you've been doing, you're going to have a tough time explaining.

A tourist in Brazil can stay for a maximum of 90 consecutive days, extendable to 180 days every one year by issuing a request at the Federal Police Department (DPF). That's not automatic; you must go to the nearest Federal Police office and fill a form and pay a fee (currently R$ 67.00 or US$ 30.70). Be prepared to present them the usual information you need when entering the country, like proof that you have enough money to cover your stay, hotel reservations etc.

All this is detailed at their Services for Foreigners area (which unfortunately has no English translation).

most European Union tourists can't extend their 90 day tourist visa (including e.g. the Netherlands, Germany, Spain, Italy, Sweden etc.), except those from the UK, Poland, Portugal and Ireland who can.