This is what I came up with:

#!/bin/perl5

use strict;
use warnings;

@ARGV = qw( file1.txt file2.txt );

my $name;
my %hash;

while (<>){
  chomp;
  my $record = $_;
  if ( $record =~ /^>/ ){
    $name = $record;
    next;
  }
  else{
    $hash{$name} .= $record
  }
}

for my $key ( keys %hash ){
  print "$key\n$hash{$key}\n"
}
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Updated: Use hash instead of a hashref