  A tree is pathwise-random if all of its paths are algorithmically random,
with a fixed upper bound on their randomness deficiency. We show that: (a) an
algorithmically random real computes a pathwise-random tree with infinitely
many paths if and only if it computes the halting problem; it follows that the
class of pathwise-random trees with unbounded width 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 positive
pathwise-random tree. We use these results to obtain models of second-order
arithmetic that separate compactness principles below weak Konigs lemma,
answering questions by Chong et al.(2019).
