  We explicitly construct a pair of immersed tori in three dimensional
Euclidean space that are related by a mean curvature preserving isometry. These
Bonnet pair tori are the first examples of compact Bonnet pairs. This resolves
a longstanding open problem on whether the metric and mean curvature function
determine a unique smooth compact immersion. Moreover, we prove these isometric
tori are real analytic. This resolves a second longstanding open problem on
whether real analyticity of the metric already determines a unique compact
immersion. Our construction uses the relationship between Bonnet pairs and
isothermic surfaces. The Bonnet pair tori arise as conformal transformations of
an isothermic torus with one family of planar curvature lines. We classify such
isothermic tori in our companion paper (arXiv:2312.14956). The above approach
stems from computational investigations of a 5x7 quad decomposition of a torus
using a discrete differential geometric analog of isothermic surfaces and
Bonnet pairs.
