  We investigate domain wall and other defect solutions in the weak-field limit
of Eddington-inspired Born-Infeld gravity as a function of $\kappa$, the only
additional parameter of the theory with respect to General Relativity. We
determine, both analytically and numerically, the internal structure of domain
walls, quantifying its dependency on $\kappa$ as well as the impact of such
dependency on the value of the tension measured by an outside observer. We find
that the pressure in the direction perpendicular to the domain wall can be, in
contrast to the weak-field limit of General Relativity, significantly greater
or smaller than zero, depending, respectively, on whether $\kappa$ is positive
or negative. We further show that the generalized von Laue condition, which
states that the average value of the perpendicular pressure is approximately
equal to zero in the weak-field limit of General Relativity, does not generally
hold in EiBI gravity not only for domain walls, but also in the case cosmic
strings and spherically symmetric particles. We argue that a violation of the
generalized von Laue condition should in general be expected in any theory of
gravity whenever geometry plays a significant role in determining the defect
structure.
