MySQL 8.0

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CREATE TABLE turip_refs (`check_id` INTEGER, `name` VARCHAR(5), `ref_no` VARCHAR(3));

INSERT INTO turip_refs (`check_id`, `name`, `ref_no`) VALUES
  ('10', 'alsak', 'xdr'),
  ('21', 'turin', 'xdr'),
  ('10', 'tend', 'ymt'),
  ('21', 'turin', 'kuy');
select ref_no
from turip_refs
where check_id in (10, 21)
group by ref_no
having count(distinct check_id) = 2


ref_no


xdr


select * 
from turip_refs
where check_id in (10, 21)
  and ref_no in (
    select ref_no
    from turip_refs
    where check_id in (10, 21)
    group by ref_no
    having count(distinct check_id) = 2  
  )


check_id
name
ref_no


10
alsak
xdr

21
turin
xdr


select check_id, name, ref_no
from (
  select *, count(*) over (partition by ref_no) counter 
  from turip_refs
  where check_id in (10, 21)
) t
where counter = 2


check_id
name
ref_no


10
alsak
xdr

21
turin
xdr