You could run a regular check that data changes are taking place, though this could be complex depending on your application.
If you have some form of audit train table that is very regularly updated (i.e. our main product has a base audit table that lists all actions that result in data being updated or deleted) then you could query that table on both servers and make sure the result you get back is the same. Something like:
SELECT CHECKSUM_AGG(*)
FROM audit_base
WHERE action_timestamp BETWEEN <time1> AND BETWEEN <time2>
where and are round values to allow for different delays in contacting the databases. For instance, if you are checking at ten past the hour you might check items from the start the last hour to the start of this hour. You now have two small values that you can transmit somewhere and compare. If they are different then something has most likely gone wrong in the replication process - have what-ever pocess does the check/comparison send you a mail and an SMS so you know to check and fix any problem that needs attention.
By using SELECT CHECKSUM_AGG(*) the amount of data for each table is very very small so the bandwidth use of the checks will be insignificant. You just need to make sure your checks are not too expensive in the load that apply to the servers, and that you don't check data that might be part of open replication transactions so might be expected to be different at that moment (hence checking the audit trail a few minutes back in time instead of now in my example) otherwise you'll get too many false alarms.
Depending on your database structure the above might be impractical. For tables that are not insert-only (no updates or deletes) within the timeframe of your check (like an audit-trail as above), working out what can safely be compared while avoiding false alarms is likely to be both complex and expensive if not actually impossible to do reliably.
You could manufacture a rolling insert-only table if you do not already have one, by having a small table (containing just an indexed timestamp column) to which you add one row regularly - this data serves no purpose other than to exist so you can check updates to the table are getting replicated. You can delete data older than your checking window, so the table shouldn't grow large. Only testing one table does not prove that all the other tables are replicating (or any other tables for that matter), but finding an error in this one table would be a good "canery" check (if this table isn't updating in the replica, then the others probably aren't either).
This sort of check has the advantage of being independent of the replication process - you are not waiting for the replication process to record exceptions in logs, you are instead proactively testing some of the actual data.