I'm assuming that readers know that we can use relative whose to refer to both people and object. I won't address this here, Fumblefingers has links to relevant posts on that subject in his answer, if you'd like to read about it.
I use the term Determiner in this piece, like the majority of modern grammars, to refer to the class of words. I use Determinative to refer to the function. Huddleston and Pullum reverse the established terminology in CGEL.
John Hanna and Fumblefingers state in comments or posts that for them whose each is grammatical. It might be the case that whose each is grammatical in the OP's intended usage for speakers of some varieties of English. I'm most familiar with so-called Standard GB, and it might be that whose each is grammatical for some other groups of speakers.
Man_From_India has suggested that phrases such as supported his each step and similar which might be grammatical. I don't think this is so. It is possible to find a few examples from Googlebooks (40 at first glance). However, these turn out mostly to be running across sentences ["his. Each step ..."], or else they are self-published unedited texts full of typos.Here's the Ngrams graph for his each step and his every stepNgrams graph for his each step and his every step. As you'll see there are no results at all for his each step:
Araucaria - Not here any more.
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Araucaria - Not here any more.
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Araucaria - Not here any more.
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