In the realm of CGI, printing to the browser is really just printing to the handle STDOUT. This means you can simply redirect STDOUT to write to both a file of your choice and the usual STDOUT.

# near the top, before you print
open FILE, '>', 'output.txt' or die ("Can't write output.txt: $!");
open STDOUT, '>&FILE>&STDOUT' or die ("Can't redirect STDOUT properly:
+ $!");
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Remember to close FILE before your script ends. (even though the script ending should cause it to go out of scope and close, better to do so explicitly).