TIMTOWTDI

instead of exporting the variable you can just check it directly in the caller's namespace. (If it even exists )

use v5.12;
use warnings;

BEGIN{
  package log;
  our $var= "default";

  sub tst {
    my $pkg= (caller)[0];
    no strict 'refs';
    say ${"${pkg}::var"}
  }

  # export
  *Other::tst = \&tst;
}


package Other;
our $var = __PACKAGE__;


{
  local $var = "foo";
  tst();
}

tst();
[download]

foo
Other
[download]

This might give your debugging interesting options...

Update

An alternative to messing with strict refs is to explicitly dig thru the stashes...

say ${$::{"${pkg}::"}{"var"}};

... Tho beauty is in the eye of the beholder ;)