India, like Italy and China, has no single dominant cuisine: Indian food comprises many different styles of cooking,
with each a product of their regional influences, from the fiery vegetarian dishes of the south to the Portuguese-influenced Goan cooking of the west, to the more familiar Mogul food of the north.
a) with each a product of their
b) with each as a product of its
c) each products of their
d) each a product of
e) each products of
Source: Official GMAT Paper Test Question
Experts please explain, why option B is wrong? Why is it wrong "with each as a product" to modify "styles of cooking" ?
Why the pronoun "its" is incorrect in option B. Isn't this pronoun referring to "each" ?
Experts please help, I am really confused.