First thing I would check is the wiring where it runs from the body to the tailgate ,through the rubber conduit. Common for broken wires here. Does the heated rear window work?

I can't think of any reason why a wiper should park in a different place depending on whether the washer has been used. But on my Pug (2008) the rear wiper is linked to reverse gear (it wipes if the main wipers are on), and occasionally the front wipers park upright for no reason, but that is only a temporary quirk as things always work normally afterwards. Must be an odd glitch in your car's brain.

Does your rear wiper always do this, or only occasionally?

The wiper motor has a worm gear with 2-3 metalic contact points so it know the position.
You should take a apart the asmbly and try to clean those contact points. Check for any cold solder joints

The wiper motor will have a set of phosphor bronze contacts which bear upon a set of corresponding metal circular contacts. Power is always applied to the motor unless it is in the park position by virtue of the fact that the circuit is completed via these rotating rings if it is in the wrong position. When the contact wears the circuit is broken and it only runs for as long as the steering column switch is in the on position ow the wash wipe circuit is active. Buy a new motor or do as I did and drill out the rivets holding it together. take an old relay and solder the contacts from this in the position of the worn contact. Reassemble with appropriate screws and it will be fine. My astra Mk4 one has been working for 7 years since it failed in this way. I initially tried a new (cheap) motor from Europarts-it lasted about 2 hours! Decided to repair the old one and bingo!