This question is related to " How to calculate characteristic classes of tensor products? " but interested in the Pontryagin classes instead of $c_1$.
Specifically, given two real vector bundles $E$, $F$ over the same base space $B$ (paracompact, or whatever you need for your answer), is there a known formula for the total Pontryagin class $p(E\otimes F)$ in terms of $p(E)$ and $p(F)$? If not, is there at least a formula for classes with small degree, analogous to the formula for $c_1$ of a tensor product of complex line bundles?
The particular example I have in mind is where $E=F$ is the realification of the tautological complex line bundle $\eta$ over $\mathbb{CP}^n$, where $n\geq 4$ so that $\eta \otimes_\mathbb{R} \eta$ has potentially non-zero $p_1$ and $p_2$.
I've checked the usual references (Milnor-Stasheff, various Hatchers) but they don't seem to tackle this particular problem. Maybe there is an argument using Chern-Weil theory and a formula for the curvature form of a tensor product of bundles with connection.