Why not just do it the simple way, as chomp will remove the '>' (if $INPUT_RECORD_SEPARATOR = '>'), then skip empty records & re-add the '>',
somthing like:

use English;

local $INPUT_RECORD_SEPARATOR = '>';

while (my $buf = <$fh>) {

       # remove '>' & skip empty records:
       chomp($buf);
       next if ($buf =~ m/^\s*?$/);
       
       my $fasta = ">$buf\n";     

};
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*UPDATE*: added use English;