Do I need a tourist visa if I am staying in the Schengen Region for more than 90 days but in each individual country for less than 90 days?

Yes, you will need a visa if you are planning to stay in the Schengen Area for more than 90 days in total, even if your stay in each individual country is less than 90 days. The Schengen rules stipulate that the cumulative duration of stay across all Schengen countries counts toward the 90-day limit within a 180-day period. For longer stays, you should apply for a national visa or residence permit specific to the country where you will be spending the most time.