Warning: I don't know if this is a correct solution. But it seems like it works, so I offer it up for criticism, just because it seems nifty:

use strict;
use warnings;

use Regexp::Common;

my $str = "a<<a> a>b<a>c<a>a";
while ($str=~/\G(?:|$RE{balanced}{-parens=>'<>'})
                ([^<]+)
                (?:|$RE{balanced}{-parens=>'<>'})/xg)
{
 # Update: Oops, this uppercases everything except what
 # the OP wanted :)
 substr($str, $-[1],$+[1]-$-[1]) = uc $1;
 pos($str) = $+[1];
}
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Update: I should read the OP again. This seems to match, but doesn't change the case like the OP wants. Nevermind (for now) :(

Update: Fixed. Though I'm still not sure if it's absolutely correct :)

Update (simplified solution below):

$str =~ s/(^|\G$RE{balanced}{-parens=>'<>'})([^<]+)/\U$1\E$2/g;
# Again? (and now I feel a DUH! coming on (:
$str =~ s/($RE{balanced}{-parens=>'<>'})/\U$1/g;
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