Is this ferry flight route possible for a B737-800 WL?
(737-800 Payload/Range chart) The winglet version has increased range compared to the chart above, per the airport planning manual.

The variant with the 31.7-tonne fuel capacity has a max ferry range of ~5,500 NM. The biggest leg in your question is ~4,400 NM, it's doable even without any tailwind.



In case the 737 in question has the basic fuel capacity (20.9 tonnes), flying eastbound usually cuts the distance due to flying in tailwind. For example, a distance of 4,400 NM at a true airspeed of 450 knots is reduced to 3,600 NM (air distance) with an average tailwind of 100 knots. And down to 3,300 NM with average tailwind of 150 knots.


  The wind may provide a head- or tailwind component, which in turn will increase or decrease the fuel consumption by increasing or decreasing the air distance to be flown.— Wikipedia


It could be down to choosing the best day to fly with the best jet streams.