I'm trying to make an overview of the study of partial hyperbolicity and there is an interesting concept of dynamical coherence which appears there. Some call it mild (see the Thesis of Pablo Carrasco, *Compact Dynamical Foliations* 2010), some call it strong and unnatural (see the work of Amy Wilkinson and Keith Burns *Dynamical coherence, accessibility and center bunching*). The definition which is the most common is that local cental-unstable $E^{cu}$ and center-stable $E^{cs}$ bundles integrate to foliations $W^{cu}$ and $W^{cs}$.
Let us suppose, that in a normally hyperbolic case, i.e. when we already have the $E^c$ that integrates to a foliation F, at which some diffeomorphism is hyperbolic.

My question is how the normally hyperbolic (i.e. partially hyperbolic on foliation) system could be dynamically incoherent and is this concept somewhat related to the concept of local product structure?

My question is, what is a simplest example of a normally hyperbolic foliation when $E^cu$ and $E^cs$ do not integrate to foliations? And how "often" does it happen in the world of normally hyperbolic foliations?

PS. Updated after a useful remark of Rafael Potrie, the definition of a dynamical coherence is now more precise.