The documentation for open provides an example of duplicating and restoring filehandles:
#!/usr/bin/perl
open(my $oldout, ">&STDOUT") or die "Can't dup STDOUT: $!";
open(OLDERR, ">&", \*STDERR) or die "Can't dup STDERR: $!";
open(STDOUT, '>', "foo.out") or die "Can't redirect STDOUT: $!";
open(STDERR, ">&STDOUT") or die "Can't dup STDOUT: $!";
select STDERR; $| = 1; # make unbuffered
select STDOUT; $| = 1; # make unbuffered
print STDOUT "stdout 1\n"; # this works for
print STDERR "stderr 1\n"; # subprocesses too
open(STDOUT, ">&", $oldout) or die "Can't dup \$oldout: $!";
open(STDERR, ">&OLDERR") or die "Can't dup OLDERR: $!";
print STDOUT "stdout 2\n";
print STDERR "stderr 2\n";