fluctuations should also be directly observable via [*in situ*]{} imaging of the spin density ($n_1-n_2$). Such a scheme was theoretically investigated for thermal spin fluctuations in Ref. [@Bisset2015]. However, in this case the quantum and thermal fluctuations will both contribute, and the temperature will need to be quite low to suppress the thermal contribution.\
[**Acknowledgments: $\,$**]{} We thank D. Baillie, T. Bienaimé, P. B. Blakie, R. Carretero-Gonz[á]{}lez, G. Ferrari, S. I. Mistakidis, A. Recati and C. Qu for useful and stimulating discussions. RNB acknowledges the EU QUIC project and Provincia Autonoma di Trento for financial support. PGK acknowledges the support of NSF-DMS-1312856 and PHY-1602994, from the ERC under FP7, Marie Curie Actions, People, International Research Staff Exchange Scheme (IRSES-605096), and the Stavros Niarchos Foundation via the Greek Diaspora Fellowship Program. CT acknowledges support by Los Alamos National Laboratory, which is operated by LANS, LLC for the NNSA of the U.S. DOE and, specifically, Contract No. DEAC52-06NA25396.

Two-component Gaussian ansatz {#Sec:AppendixAna}
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To gain a theoretical handle on the miscibility transition in the quasi-2D setting we apply a chirped Gaussian ansatz (superscript $\mathcal V$) as an approximation to the wavefunctions (similar to what was done in 1D by [@Navarro2009]), i.e., $$\psi^{\mathcal 