  We establish a surface order large deviation estimate for the magnetisation
of low temperature $\phi^4_3$. As a byproduct, we obtain a decay of spectral
gap for its Glauber dynamics given by the $\phi^4_3$ singular stochastic PDE.
Our main technical contributions are contour bounds for $\phi^4_3$, which
extends 2D results by Glimm, Jaffe, and Spencer (1975). We adapt an argument by
Bodineau, Velenik, and Ioffe (2000) to use these contour bounds to study phase
segregation. The main challenge to obtain the contour bounds is to handle the
ultraviolet divergences of $\phi^4_3$ whilst preserving the structure of the
low temperature potential. To do this, we build on the variational approach to
ultraviolet stability for $\phi^4_3$ developed recently by Barashkov and
Gubinelli (2019).
