A *Canadian* tree (also called a *weak Kurepa* tree) is a tree of height $\omega_1$ with levels of cardinality $\leq \omega_1$ and at least $\omega_2$ many uncountable branches. Let's call a Canadian tree *thick* if it has $2^{\omega_1}$ many uncountable branches. Thick Canadian trees of course exist under CH (just take the complete binary tree of height $\omega_1$).

QUESTION: Is the existence of a thick Canadian tree consistent with $\omega_1 < 2^\omega < 2^{\omega_1}$?

As noted by Hannes Jakob in the comments, to get even a thick Kurepa tree consistently with $\omega_1 < 2^\omega=2^{\omega_1}$ it's enough to add $\omega_2$ many Cohen reals to a model of GCH+There is a Kurepa tree.