  Double-stripe magnetism $[\mathbf{Q}=(\pi/2, \pi/2)]$ has been proposed as
the magnetic ground state for both the iron-telluride and BaTi$_2$Sb$_2$O
families of superconductors. Double-stripe order is captured within a
$J_1-J_2-J_3$ Heisenberg model in the regime $J_3 \gg J_2 \gg J_1$.
Intriguingly, besides breaking spin-rotational symmetry, the ground state
manifold has three additional Ising degrees of freedom associated with
bond-ordering. Via their coupling to the lattice, they give rise to an
orthorhombic distortion and to two non-uniform lattice distortions with
wave-vector $(\pi, \pi)$. Because the ground state is four-fold degenerate,
modulo rotations in spin space, only two of these Ising bond order parameters
are independent. Here we introduce an effective field theory to treat all Ising
order parameters, as well as magnetic order, and solve it within a large-$N$
limit. All three transitions, corresponding to the condensations of two Ising
bond order parameters and one magnetic order parameter are simultaneous and
first order in three dimensions, but lower dimensionality, or equivalently
weaker interlayer coupling, and weaker magnetoelastic coupling can split the
three transitions, and in some cases allows for two separate Ising phase
transitions above the magnetic one.
