  We define a birational map between labelings of a rectangular poset and its
associated trapezoidal poset. This map tropicalizes to a bijection between the
plane partitions of these posets of fixed height, giving a new bijective proof
of a result by Proctor. We also show that this map is equivariant with respect
to birational rowmotion, resolving a conjecture of Williams and implying that
birational rowmotion on trapezoidal posets has finite order.
