Are there records of monozygotic twins in which one experiences androgen insensitivity syndrome
This is an important question to geneticists because monozygotic twins are the only ethically acceptable way to research genetically identical humans.  

I'm sure the sort of non-symmetrical fission where specific DNA coding for genes are not present or are modified to completely different phenotype in the formation of the twins probably has happened, but it incredibly rare.  In stem cells, asymmetric division has been observed frequently (warning: google doesn't like PlosOne right now).  In development, unless its actively been activated as an asymmetric division, this would be incredibly rare- this would be akin to a birth defect in a single fetus - how many cases of mixed male and female cells in a single baby. I can't even find a reference for such a thing.  Much more common are semi-identical twins, where two different sperm fertilize a single egg which may have split.  Semi-twins are still vanishingly rare - the first report was in 2007). 


  Such twins are probably very rare. Their existence and discovery relies on three unusual, and possibly unlinked, events: first, that an egg fertilized by two sperm develops into a viable embryo; second, that this embryo splits to form twins; and third, that the children come to the attention of science.


This might be the case you are thinking of?  In one case reported there was a gender difference.  


  Souter and her colleagues investigated the twins' genetic makeup because one was born with ambiguous genitalia. This twin turned out to be a 'true hermaphrodite', with both ovarian and testicular tissue. The other twin is anatomically male.


They weren't monozygotic though.  Just others from the same mother ;)

Monozygotic twins are not completely identical, epigenetic fingerprinting - including DNA methylation and chromatin DNA binding patterns (pnas - paywall) change while the fetus develops and as the child and adult age.  This can cause quite significant differences as the twins age. Epigenetic markers change the dynamics of brain development and can make a difference in the occurance of leukemia.  

There are also modifications to the DNA sequences of cells, copy number variations (CNVs) are multiple (sometimes hundreds or thousands of times over) repeats of short (2+ base pairs) DNA sequences.  These also occur as the cells differentiate and divide to form tissues and organs.  Twins show significant differences there too as this is not a deterministic mechanism in development. 

I can't find any references to twins showing asymmetric AIS. but the number of variants requiring this makes it sounds like its possible.  If you have a reference post it and I can have a look? 

note added: this paper shows that methylation profiles appear identical at least in utero.