[TYPO3] RealURL - alias domains

[Patrick Fitz-Simons]

G'day folks
(Tipping hat to Dimitry and his excellent endeavors of programming and 
documentation)

I believe I have missed a config point of RealURL in a recent deployment 
  of typo3 4.1.2

1 web site with
	2 domains .. 1 main , 1 alias


observations:

1) visitor lands on BaseURL "A.com"
	- menu urls are generated in "A.com"
GOOD!
live example www.seejudy.com

2) visitor lands on alias url "B.com"
	- menu urls are also generated in "A.com"
NOT SO GOOD!
live example www.judyweinstock.com


I would like for both sites to resolve and navigate discreetly.

Using RealURL, is it possible to "have 2 baseURL's"?
or in other words
Can  both domains (the main (baseURL) & alias), resolve and keep 
generated urls, links and menu item paths within the same domain that 
the vistor lands?..

Any clues would be appropriately appreciated
as I have searched the world over and feel I have missed the point or am 
lacking the proper vocabulary or question..

Many thank in advance!
Patrick Fitz-Simons
http://www.patrick.fitz-simons.org