  The constraints imposed on heterotic compactifications by global consistency
and phenomenology seem to be very finely balanced. We show that weakening these
constraints, as was proposed in some recent works, is likely to lead to
frivolous results. In particular, we construct an infinite set of such
frivolous models having precisely the massless spectrum of the MSSM and other
quasi-realistic features. Only one model in this infinite collection (the one
constructed in arXiv:hep-th/0512149) is globally consistent and supersymmetric.
The others might be interpreted as being anomalous, or as non-supersymmetric
models, or as local models that cannot be embedded in a global one. We also
show that the strongly coupled model of arXiv:hep-th/0512149 can be modified to
a perturbative solution with stable SU(4) or SU(5) bundles in the hidden
sector. We finally propose a detailed exploration of heterotic vacua involving
bundles on Calabi-Yau threefolds with Z_6 Wilson lines; we obtain many more
frivolous solutions, but none that are globally consistent and supersymmetric
at the string scale.
