some way to do it that was native to Perl.

The "native" way: with a closure:

my $variable = 3;

my $string = sub { return "Variable: $variable\n"; };
print &$string;

$variable = 4;
print &$string;
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I assume here that by "native" you mean "without using a module". However, be advised that many modules are written using "native" pure perl and that modules are as much a part of the language as the print function.