  A tree is pathwise-random if all of its paths are Martin-Lof random. We show
that (a) no weakly 2-random real computes a perfect pathwise-random tree; it
follows that the class of perfect pathwise-random trees is null, with respect
to any computable measure; (b) there exists a positive-measure pathwise-random
tree which does not compute any complete extension of Peano arithmetic; and (c)
there exists a perfect pathwise-random tree which does not compute any tree of
positive measure and finite randomness deficiency. We then obtain models of
second-order arithmetic that separate compactness principles below weak Konigs
lemma, answering questions by Chong et al.(2019).
