  We investigate the asymptotic distribution of the profile likelihood ratio
(PLR) when constraining effective field theories (EFTs) and show that Wilks'
theorem is often violated, meaning that we should not assume the PLR to follow
a $\chi^2$-distribution. We derive the correct asymptotic distributions when
either one or two real EFT couplings modulate observable cross sections with a
purely linear or quadratic dependence. We then discover that when both the
linear and quadratic terms contribute, the PLR distribution does not have a
simple form. In this case we provide a partly-numerical solution for the
one-parameter case. Using a novel approach, we find that the constants which
define our asymptotic distributions may be obtained experimentally using a
profile of the Asimov likelihood contour. Our results may be immediately used
to obtain the correct coverage when deriving real-world EFT constraints using
the PLR as a test-statistic.
