  Integrable field theories in two dimensions are known to originate as defect
theories of 4d Chern-Simons and as symmetry reductions of the 4d anti-self-dual
Yang-Mills equations. Based on ideas of Costello, it has been proposed in work
of Bittleston and Skinner that these two approaches can be unified starting
from holomorphic Chern-Simons in 6 dimensions. We provide the first complete
description of this diamond of integrable theories for a family of deformed
sigma models, going beyond the Dirichlet boundary conditions that have been
considered thus far. Starting from 6d holomorphic Chern-Simons theory on
twistor space with a particular meromorphic 3-form $\Omega$, we construct the
defect theory to find a novel 4d integrable field theory, whose equations of
motion can be recast as the 4d anti-self-dual Yang-Mills equations. Symmetry
reducing, we find a multi-parameter 2d integrable model, which specialises to
the $\lambda$-deformation at a certain point in parameter space. The same model
is recovered by first symmetry reducing, to give 4d Chern-Simons with
generalised boundary conditions, and then constructing the defect theory.
