What makes you think the BEGIN treats %hash as if it were a file-scoped global? Try moving the my %hash; *after* the begin block.

Note also that your claim the keys are retained doesn't seem to be true:

use strict;
use warnings;
use 5.010;

sub routine {
    my %hash;
    BEGIN {
        $hash{KEY} = "string";
    }
    $hash{FOO} = "something else";
    say join " ", keys %hash;
}

routine; routine; routine;

__END__
Variable "%hash" will not stay shared at /tmp/hh line 12.
KEY FOO
FOO
FOO
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