  We introduce and study new interacting topological states that arise in
time-reversal symmetric bands with an underlying obstruction to forming
localized states. If the $U(1)$ valley symmetry linked to independent charge
conservation in each time-reversal sector is spontaneously broken, the
corresponding `excitonic' order parameter is forced to form a topologically
non-trivial texture across the Brillouin zone. We show that the resulting
phase, which we dub a textured exciton insulator, cannot be given a
local-moment description due to a form of delicate topology. Using toy models
of bands with Chern or Euler obstructions to localization we construct explicit
examples of the Chern or Euler texture insulators (CTIs or ETIs) they support,
and demonstrate that these are generically competitive ground states at
intermediate coupling. We construct field theories that capture the response
properties of these new states. Finally, we identify the incommensurate
Kekul\'e spiral phase observed in magic-angle bi- and trilayer graphene as a
concrete realization of an ETI.
