  We consider a model of directed polymers on regular trees with complex-valued
random weights introduced by Cook and Derrida [CD90] and studied mathematically
by Derrida, Evans and Speer [DES93]. In addition to the usual weak-disorder and
strong-disorder regimes, the phase diagram of the model contains a third region
due to the effects of the random phases that cannot be observed in the model
with positive weights.
  In this work, we extend the results of [DES93] in two directions. First, we
remove the hypothesis on the independence of the random radii and random phases
of the environment from most of the phase diagram, and second, under mild
assumptions on the law of the environment, we strengthen the convergence of the
free energy (shown to hold in probability in [DES93]) to an almost sure
convergence. Along the way, many of the arguments of [DES93] are simplified.
