Interesting new development: The keep options as described above works only if the cells themselves don't have any keep options. If I set the first cell to stay with next cell in the first table, then the table doesn't stay with the second table, the first table remains on the first page.
If the cells don't have to stay together, then the second cell in the first table does get pulled to the second page.
Of course what I need is for both cells to move together to the page with the second table.