Thanks, everyone, for your help. My travel agent is working with the AA RTW desk, and everything seems to be working out. I have purchased, for US$8519 (including taxes), a RTW ticket with the following itinerary:

BOS-LHR
LHR-SIN
SIN-BKK
BKK-HKG
HKG-JFK
JFK-LAX
LAX-ORD
ORD-BOS

Next week, I'll just use the BOS-LHR-SIN-BKK portion, and will purchase a second RTW ticket in BKK with the following itinerary:

BKK-HKG
HKG-JFK
JFK-BOS
JFK-SFO
SFO-BOS
BOS-PBI
PBI-BOS
BOS-LHR
LHR-MAD
MAD-LHR
LHR-BKK

This prices out to just a little more than I already was paying to travel in F from BOS-LHR-SIN-HKG-JFK-BOS, and I get a second trip to Asia, as well as a number of extra U.S. trips to the West Coast.