  Recent experimental data on Bose-Einstein Condensation (BEC) of magnons in
the spin-gap compound Yb$_2$Si$_2$O$_7$ revealed an asymmetric BEC dome
arXiv:1810.13096v2. We examine modifications to the Heisenberg model on a
breathing honeycomb lattice, showing that this physics can be explained by
competing forms of weak anisotropy. We employ a gamut of analytical and
numerical techniques to show that the anisotropy yields a field driven phase
transition from a state with broken Ising symmetry to a phase which breaks no
symmetries and crosses over to the polarized limit.
