I imagine this is fairly elementary, but I couldn't find a good reference.
Let $G$ be a compact Lie group and $\pi:P\rightarrow M$ a principal $G$-bundle over a (compact) manifold $M$. Let $F$ be another (compact) manifold on which $G$ acts on the left through a smooth map $\rho:G\rightarrow Diff(F)$. We form the balanced product $P\times_{\rho}F$ which admits a smooth submersion onto $M$ with fibre $F$.
Is there a nice global description of the tangent bundle $T(P\times_\rho F)\rightarrow P\times_\rho F$?
Locally the question is pretty clear, since the local triviality of $P$ transfers to $P\times_\rho F$, to give smooth fibrewise local trivialisations $(P\times_\rho F)|_U\cong U\times F$ for suitable open $U\subseteq M$. Thus locally $T(P\times_\rho F)$ looks like $TU\times TF$, and it seems to me that globally it should be the quotient vector bundle $TP\times_{T\rho}TF$ formed as the balanced product by the induced tangent representation $T\rho$ of $G$.