  We extend the periodicity of birational rowmotion for rectangular posets to
the case when the base field is replaced by a noncommutative ring (under
appropriate conditions). This resolves a conjecture from 2014. The proof uses a
novel approach and is fully self-contained.
  Consider labellings of a finite poset $P$ by $\left|P\right| + 2$ elements of
a ring $\mathbb{K}$: one label associated with each poset element and two
constant labels for the added top and bottom elements in $\hat{P}$. *Birational
rowmotion* is a partial map on such labellings. It was originally defined by
Einstein and Propp for $\mathbb{K}=\mathbb{R}$ as a lifting (via
detropicalization) of *piecewise-linear rowmotion*, a map on the order polytope
$\mathcal{O}(P) := \{\text{order-preserving } f: P \to[0,1]\}$. The latter, in
turn, extends the well-studied rowmotion map on the set of order ideals (or
more properly, the set of order filters) of $P$, which correspond to the
vertices of $\mathcal{O}(P)$. Dynamical properties of these combinatorial maps
sometimes (but not always) extend to the birational level, while results proven
at the birational level always imply their combinatorial counterparts. Allowing
$\mathbb{K}$ to be noncommutative, we generalize the birational level even
further, and some properties are in fact lost at this step.
  In 2014, the authors gave the first proof of periodicity for birational
rowmotion on rectangular posets (when $P$ is a product of two chains) for
$\mathbb{K}$ a field, and conjectured that it survives (in an appropriately
twisted form) in the noncommutative case. In this paper, we prove this
noncommutative periodicity and a concomitant antipodal reciprocity formula. We
end with some conjectures about periodicity for other posets, and the question
of whether our results can be extended to (noncommutative) semirings.
