Steve -- depends on the lathe -- some come with good live centers- Most come without 60 degree tips in the live centers (Noval has that with the live center and Oneway/Powermatic live centers have a pen turners tip that can be easily added).
Matthew -- Did you use your 60 degree live center tip??? If you did not, and have not been using a 60 degree live center - those tips get distorted easily. Some times that get mangled twisted and mutilated.
Most american/european market lathes are aligned against the inside of the ways. some tailstocks have a fair amount of play and will be in better alignment if pushed or pulled against the ways before locking it down. Woodlathes do not have the need to be in as tight an alignment as metal lathes for the most part -- mandrel turning being an application where better alignment is mutch better.
Shimming that much will not be easy as the shims need to be done so the axis of the spindle matches the axis if the tailstock. (shimming one end of the headstock to make the point align, but having the spindle pointed to the cieling is not going to be effective,)
Or you can move to turning between centers which is much more tolerant of alignment because the dead/live centers can function as "universal joints" for minor alignment misses. Fits at the end of the blanks becomes very very important if there are alignment anomolies as you do not want the blank-bushing connection to be part of the "universal joint" movement.
some of the big lathes with moving heads and beds have can be a challenge to get back into alignment. My big lathe has a movable bed and can be 5-10 thousand of an inch out of perfect alignment easily. Your 1.5mm (or 3/ 32 inch or 0.090 inch) is more than desirable, and is comperable to lathes of many years ago --
Make and Model?????