Just before giving up I found it!

I needed to limit both '.*' to the smallest match '.*?':

$body =~ /<tag1>\[(\w+)\]<\/tag1>.*?<tag2>(.*?)<\/tag2>/gs

With the '?' pattern matching quantifier in combination with 's' modifier it will get strings with multiple lines

Took me a few hours but that's the r.e. learning curve ;-)