  We investigate approximate emergent nonabelian symmetry in a class of weakly
first order `deconfined' phase transitions using Monte Carlo simulations and a
renormalization group analysis. We study a transition in a 3D classical loop
model that is analogous to a deconfined 2+1D quantum phase transition in a
magnet with reduced lattice symmetry. The transition is between the N\'eel
phase and a twofold degenerate valence bond solid (lattice-symmetry-breaking)
phase. The combined order parameter at the transition is effectively a
four-component superspin. It has been argued that in some weakly first order
`pseudocritical' deconfined phase transitions, the renormalization group flow
can take the system very close to the ordered fixed point of the symmetric
$O(N)$ sigma model, where $N$ is the total number of `soft' order parameter
components, despite the fact that $O(N)$ is not a microscopic symmetry. This
yields a first order transition with unconventional phenomenology. We argue
that this occurs in the present model, with $N=4$. This means that there is a
regime of lengthscales in which the transition resembles a `spin-flop'
transition in the ordered $O(4)$ sigma model. We give numerical evidence for
(i) the first order nature of the transition, (ii) the emergence of $O(4)$
symmetry to an accurate approximation, and (iii) the existence of a regime in
which the emergent $O(4)$ is `spontaneously broken', with distinctive features
in the order parameter probability distribution. These results may be relevant
for other models studied in the literature, including 2+1D QED with two
flavours, the `easy-plane' deconfined critical point, and the N\'eel--VBS
transition on the rectangular lattice.
