No. Coherence means "fixed, definite phase relation", or in a polychromatic context it can mean "definite phase relation at any given frequency". What would be the phase relation of the horizontally- and vertically-polarized waves? $0^\circ$? $90^\circ$? There's no reason for any definite phase relation.
Here's a more specific way to think about it. Unpolarized light is exactly the same as light where 50% of the photons are clockwise-circularly-polarized and 50% are couterclockwise-circularly-polarized. For the former, the vertical polarization component is $+90^\circ$ out-of-phase with the horizontal component; for the latter it's $-90^\circ$. These two parts would lead to equal and opposite opposite interference fringes; altogether, there are no interference fringes.
If the light from the source is initially fully- or partially-polarized, then it is definitely possible to get interference fringes in the way you describe.