I don't know specifics about UA RTW reward, but in general it is better value to book
3 round trips with stop-overs on each, to create 2 RTW's, for approx the same points
as the RTW reward by itself. I am presuming that the UA RTW reward is similar to
most others in restricting you to 5 stop-overs ( not like a revenue RTW with 15 allowed).

Better value:

Ticket one: North America - 1st stopover Europe - Destination Europe/
Ticket two : Europe - 1st Stopover Asia - Destination Asia
Ticket three : Asia - 1st Stopover North America - North America ( home - take a break for a few months) then:

Ticket three - North America - stopover Asia - Starting point Asia
Ticket two - Asia - stopover Europe/Asia - Starting Point Europe
Ticket one: Europe/ stop-over Europe/North America - home

Of couse variation possible to South America/Africa/Australia, etc.