# Define them as hash references.
my $RECORDxRI_count_REF = {};
my $EACH_RECORD_REF = {};

while (<DATA>) {
    ....
    # Call build list--now you don't need to
    # take references to the hashes, because they're
    # already references
    build_list( $RECORD, $RI, $RECORDxRI, $RECORDxRI_count_REF, $EACH_RECORD_REF );
    ....
}

sub build_list {
    my( $RECORD, $RI, $RECORDxRI, $RECORDxRI_count_REF, $EACH_RECORD_REF ) = @_;

    # These are hash references. To access them,
    # you *don't* need to assign them to hashes.
    # Just use the -> operator, which automatically
    # dereferences them.
    $RECORDxRI_count->{$RECORDxRI}++;
    $EACH_RECORD->{$RECORD} = '';
}