There more than one kind of strict errors. Undeclared variables will lead to strict errors. my and our help fix these errors. However, the error in include_file.pl is not caused by undeclared variables. It's caused by using symbolic references. It's possible to write code that fetches a reference to a symbol while being strict-friendly, but it's much simple to temporarily disable checks for symbolic references. That can be done as follows:

# include_file.pl

# Pulled from the DB.
my $test          = "my_variable_name";
my $default_value = "the value I set";

my $var_ref = do { no strict 'refs'; \${$test} };

$$var_ref = $default_value;
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By the way, script.pl doesn't compile. I'm going to assume your real script.pl doesn't have that problem.