Postgresql synchronous_commit off and select queries
Have the sync program update a counter or timestamp in some dummy table and commit the update:

begin;
select * from async_inserted;
-- maybe do something in the external system here
update dummy set synctime=now();
commit;
-- or maybe do something external here instead


Of course if the update or commit fails, that could still leave you not knowing what state the external system should be in.  You might need to use prepared transactions (aka "2 phase commit") to bond the two systems together atomically.