  We identify the unique stress tensor deformation which preserves
zero-birefringence conditions in non-linear electrodynamics, which is a $4d$
version of the ${T\overline{T}}$ operator. We study the flows driven by this
operator in the three Lagrangian theories without birefringence -- Born-Infeld,
Plebanski, and reverse Born-Infeld -- all of which admit ModMax-like
generalizations using a root-${T\overline{T}}$-like flow that we analyse in our
paper. We demonstrate one way of making this root-${T\overline{T}}$-like flow
manifestly supersymmetric by writing the deforming operator in $\mathcal{N} =
1$ superspace and exhibit two examples of superspace flows. We present scalar
analogues in $d = 2$ with similar properties as these theories of
electrodynamics in $d = 4$. Surprisingly, the Plebanski-type theories are fixed
points of the classical ${T\overline{T}}$-like flows, while the
Born-Infeld-type examples satisfy new flow equations driven by relevant
operators constructed from the stress tensor. Finally, we prove that any theory
obtained from a classical stress-tensor-squared deformation of a conformal
field theory gives rise to a related ``subtracted'' theory for which the
stress-tensor-squared operator is a constant.
