What is happening is that a multi value field actually creates a hidden child table, that is how it does the one to many. The link to this table can get lost depending on how you deploy the split db. In the split db delete all linked tables and then relink. You can try just deleting the table with the MV field and relinking, but from what I have read most people need to delete and relink all. If this happens a lot there is code you can find to delete and relink.
You should be able to verify this. On the standalone when you look at a table with the MV field you should see a + sign on those records that will show the subdatasheet. My guess on the split db there will be no subdatasheet.