If i remember correctly, a disavantage of the tan-half link is that is can make transform an unimodal distribution on the constrained circular space into a (potentially strongly) bimodal distribution in the unconstrained space, if there is relevant posterior mass near the “cutting point” of the link function in constrained space. Even if there is negligible mass there I imagine the sampler could probably get stuck on the wrong side of that edge if there’s a strong gradient there.
Another way to parameterise a circular distribution is to use vectors of higher dimension projected down to the desired dimensionality. This avoids both the bimodality and the unboundedness issue. See e.g. this thread for some discussion about some important technicalities behind that: