Replace newlines only if not inside braces

[jbryan]

Regex gurus: How do I NOT match something inside something else? E.g. I want to match X but not if its in braces {X}?

Example:

I want to replace all newlines (\n) with (\n<br>), but not if they are inside double-braces. For example:

my $data = q|
foo
bar
{{
alpha
beta
}}
baz
|;

In $data, above, the lines with 'foo' and 'bar' should have the <br> added, but the lines with 'alpha' and 'beta' should not be touched.

Am I just missing something simple in perlre that would make this work? I don't even know where to start - other than some non-regex solution entirely.