Limitting full join to one match

Dear Experts,

I have a couple of tables that I want to reconcile, finding rows
that match and places where rows are missing from one table or the
other:

db=> select * from a;
+------------+--------+
| date | amount |
+------------+--------+
| 2018-01-01 | 10.00 |
| 2018-02-01 | 5.00 | <-- missing from b
| 2018-04-01 | 5.00 |
+------------+--------+

db=> select * from b;
+------------+--------+
| date | amount |
+------------+--------+
| 2018-01-01 | 10.00 |
| 2018-03-01 | 8.00 | <-- missing from a
| 2018-04-01 | 5.00 |
+------------+--------+

db=> select a.date, a.amount, b.date, b.amount from a full join b using (date,amount);
+------------+--------+------------+--------+
| date | amount | date | amount |
+------------+--------+------------+--------+
| 2018-01-01 | 10.00 | 2018-01-01 | 10.00 |
| 2018-02-01 | 5.00 | | |
| | | 2018-03-01 | 8.00 |
| 2018-04-01 | 5.00 | 2018-04-01 | 5.00 |
+------------+--------+------------+--------+

This works fine until I have multiple items with the same date
and amount:

db=> select * from a;
+------------+--------+
| date | amount |
+------------+--------+
| 2018-01-01 | 10.00 |
| 2018-02-01 | 5.00 |
| 2018-04-01 | 5.00 |
| 2018-05-01 | 20.00 | <--
| 2018-05-01 | 20.00 | <--
+------------+--------+

db=> select * from b;
+------------+--------+
| date | amount |
+------------+--------+
| 2018-01-01 | 10.00 |
| 2018-03-01 | 8.00 |
| 2018-04-01 | 5.00 |
| 2018-05-01 | 20.00 | <--
| 2018-05-01 | 20.00 | <--
+------------+--------+

db=> select a.date, a.amount, b.date, b.amount from a full join b using (date,amount);
+------------+--------+------------+--------+
| date | amount | date | amount |
+------------+--------+------------+--------+
| 2018-01-01 | 10.00 | 2018-01-01 | 10.00 |
| 2018-02-01 | 5.00 | | |
| | | 2018-03-01 | 8.00 |
| 2018-04-01 | 5.00 | 2018-04-01 | 5.00 |
| 2018-05-01 | 20.00 | 2018-05-01 | 20.00 | 1
| 2018-05-01 | 20.00 | 2018-05-01 | 20.00 | 2
| 2018-05-01 | 20.00 | 2018-05-01 | 20.00 | 3
| 2018-05-01 | 20.00 | 2018-05-01 | 20.00 | 4
+------------+--------+------------+--------+

It has, of course, put four rows in the output for the new items.

So my question is: how can I modify my query to output only two rows,
like this:?

+------------+--------+------------+--------+
| date | amount | date | amount |
+------------+--------+------------+--------+
| 2018-01-01 | 10.00 | 2018-01-01 | 10.00 |
| 2018-02-01 | 5.00 | | |
| | | 2018-03-01 | 8.00 |
| 2018-04-01 | 5.00 | 2018-04-01 | 5.00 |
| 2018-05-01 | 20.00 | 2018-05-01 | 20.00 | 1
| 2018-05-01 | 20.00 | 2018-05-01 | 20.00 | 2
+------------+--------+------------+--------+

Any suggestions anyone?

The best I have found so far is something involving EXCEPT ALL:

db=> select * from a except all select * from b;
db=> select * from b except all select * from a;

That's not ideal, though, as what I ultimately want is something
that lists everything with its status:

+------------+--------+--------+
| date | amount | status |
+------------+--------+--------+
| 2018-01-01 | 10.00 | OK |
| 2018-02-01 | 5.00 | a_only |
| 2018-03-01 | 8.00 | b_only |
| 2018-04-01 | 5.00 | OK |
| 2018-05-01 | 20.00 | OK |
| 2018-05-01 | 20.00 | OK |
+------------+--------+--------+

That would be easy enough to achieve from the JOIN.

Thanks, Phil.