We study three Markov processes on infinite, unrooted, regular trees: the stochastic Ising model (also known as the Glauber heat bath dynamics of the Ising model), a majority voter dynamic, and a coalescing particle model. In each of the three cases the tree exhibits a preferred direction encoded into the model. For all three models, our main result is the existence of a stationary but non-reversible measure. For the Ising model, this requires imposing that the inverse temperature is large and choosing suitable non-uniform couplings, and our theorem implies the existence of a stationary measure which looks nothing like a low-temperature Gibbs measure. The interesting aspect of our results lies in the fact that the analogous processes do not have non-Gibbsian stationary measures on $\mathbb Z^d$, owing to the amenability of that graph. In fact, no example of a stochastic Ising model with a non-reversible stationary state was known to date.