  Scalar effective field theories with enhanced soft limits behave in many ways
like gauge theories and gravity. In particular, symmetries fix the structure of
interactions and the tree-level S-matrix in both types of theories. We explore
how this analogy persists in the presence of matter by considering theories
with additional fields coupled to the Dirac-Born-Infeld (DBI) scalar or the
special galileon in a way that is consistent with their symmetries. Using
purely on-shell arguments, we show that these theories obey analogues of the
S-matrix equivalence principle whereby all matter fields must couple to the DBI
scalar or the special galileon through a particular quartic vertex with a
universal coupling. These equivalence principles imply the universality of the
leading double soft theorems in these theories, which are scalar analogues of
Weinberg's gravitational soft theorem, and can be used to rule out interactions
with massless higher-spin fields when combined with analogues of the
generalized Weinberg-Witten theorem. We verify in several examples that
amplitudes with external matter fields nontrivially exhibit enhanced single
soft limits and we show that such amplitudes can be constructed using soft
recursion relations when they have sufficiently many external DBI or special
galileon legs, including amplitudes with massive higher-spin fields. As part of
our analysis we construct a recently conjectured special galileon-vector
effective field theory.
