I'm not at all sure there is an 'derr obvious' answer to your problem - right on the contrary.

I think you'll have to go another way; consider for example changing your programs to explicitely use

use SHARED_INTEGERS;

my $calculation = $SHARED_INTEGERS::alfa * $SHARED_INTEGERS::beta;
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where you'd have an module named SHARED_INTEGERS.pm that would export _nothing_ but might look like

package SHARED_INTEGERS.pm

$alfa = 123;
$beta = 456;
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I know it's rather in the opposite direction that you are looking for - still, maybe it's worth thinking once more about the pros and cons of the different approaches.
Maybe you could also consider the constant pragma.