Keep rows only if Date exists in both table
If you want to perform a DELETE action:

You'll delete all rows in table1 not present in table2 (by date field) and viceversa

DELETE FROM table1
WHERE NOT EXISTS(
    SELECT 'X'
    FROM table2
    WHERE table1.date = table2.date
)


and

DELETE FROM table2
WHERE NOT EXISTS(
    SELECT 'X'
    FROM table1
    WHERE table2.date = table1.date
)


If you want to perform a SELECT action

You'll show only rows presents in table1 and table2

SELECT * FROM table1
WHERE EXISTS(
    SELECT 'X'
    FROM table2
    WHERE table1.date = table2.date
)
UNION ALL
SELECT * FROM table2
WHERE EXISTS(
    SELECT 'X'
    FROM table1
    WHERE table2.date = table1.date
)