Aaron, the requests are not through multiple ASM, one VIP per single ASM policy. see an obfuscated session seen through HTTP watch for a particular URL below. As you can see, there are indeed 3 unique TS Cookies names, these names then in turn generate 5 unique values
Cooke NameDirectionValuePathDomainExpire
CookradSentF/www.foobar.comSat, 12-Dec-2015 00:00:00 GMT
CookediSentpriv/www.foobar.comSat, 12-Dec-2015 00:00:00 GMT
cooklanguageSentfr/www.foobar.comSat, 12-Dec-2015 00:00:00 GMT
regionSentUS/www.foobar.comSat, 12-Dec-2015 00:00:00 GMT
COOKIE-IDSent18670353/confidential/www.foobar.com(Session)
COOKIE-IDReceived15149633/confidentialwww.foobar.com(Session)
CookSSent236dea034e8c2723ea/confidential/www.foobar.com(Session)
CookSReceived236dea034e8c2723ea/confidentialwww.foobar.com(Session)
JSESSIONIDSent0000nh2UAvelywwm0F9q:j/1/confidential/www.foobar.com(Session)
JSESSIONIDReceived0000nh2UAvelywwm0F9q:j/1/confidentialwww.foobar.com(Session)
pcook01Sent3567549882.45573.0220/www.foobar.com(Session)
TSb5519bSentad0a3a8332ce4b70d54a7504cfd7fbf2663a5ac01ed1c0b1494248d0f945da7s14b88852364a514e01649925ef0fe8cac9s2/www.foobar.com(Session)
TSb5519bReceivedc47b3230e29e90bcc2f4207a8b391ec5663a5ac01ed1c0b149424a12f945da7s14b88852364a514e01649925ef0fe8cac9s2/www.foobar.com(Session)
TSd4a7b3Sent62018140da31d410d46a211258255069663a5ac01ed1c0b1494248d0f945da7s14b88852364a514e01649925ef099d595fs2d626fe9dd32c60ed/confidential/www.foobar.com(Session)
TSfb4472Sent7b94953a2a07baf46135e2e4cf5909b4663a5ac01ed1c0b1494248d0f945da7s14b88852364a514e01649925ef099d595fs260ac0ec50efbd2580aaf8394b766037cagt6/confidential/www.foobar.com(Session)
TSfb4472Received57592a963d1114d054522ca088ceceb3663a5ac01ed1c0b1494248d0f945da7s14b88852364a514e01649925ef099d595fs260ac0ec50efbd2580aaf83943e858adeagt6/confidentialwww.foobar.com(Session)