  A central tenant in the classification of phases is that boundary conditions
cannot affect the bulk properties of a system. In this work, we show striking,
yet puzzling, evidence of a clear violation of this assumption. We use the
prototypical example of an XYZ chain with no external field in a ring geometry
with an odd number of sites and both ferromagnetic and antiferromagnetic
interactions. In such a setting, even at finite sizes, we are able to calculate
directly the spontaneous magnetizations that are traditionally used as order
parameters to characterize the system's phases. When ferromagnetic interactions
dominate, we recover magnetizations that in the thermodynamic limit lose any
knowledge about the boundary conditions and are in complete agreement with
standard expectations. On the contrary, when the system is governed by
antiferromagnetic interactions, the magnetizations decay algebraically to zero
with the system size and are not staggered, despite the AFM coupling. We term
this behavior {\it ferromagnetic mesoscopic magnetization}. Hence, in the
antiferromagnetic regime, our results show an unexpected dependence of a local,
one--spin expectation values on the boundary conditions, which is in contrast
with predictions from the general theory.
