There have been discussions before calling for reliable and accurate alternatives to GPS. The Vikings may have had the answer without knowing the physics. They were adept navigators at latitudes where frequent cloudiness and low sun angle(winter) could have made direct observation of the sun difficult. Legend has it that they had a magic stone, called a Sunstone, that allowed them to navigate by the sun even after dark. You're wondering how maybe, and I'm wondering when will this be available on a chip.
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Icelandic Spar, Calcite is probably available on a gem and mineral site. It's not like a diamond, nor is it the only mineral that exhibits double refraction.
The article has a link to a slightly better explanation of the birefringence that calcite causes. The light gets split into 2 beams and when they converge as the crystal is rotated, that indicates the azimuth of the polarized light source, the sun. GPS provides a triangulated position fix, using the minute differences in time it takes for multiple satellite signals to reach the Rx. Only knowing the azimuth of the sun isnt quite a position fix, but with the way sensor fusion is possible, a chip which determines the azimuth of the sun based on light properties could provide one piece of a fused solution.
I'm no engineer or anything, but I know it would take some serious technology to package that capability.
Hmm...next project!? If only I could get one, anythings possible with an arduino!