  We construct an effective field theory of a two-neutron halo nucleus in the
limit where the two-neutron separation energy $B$ and the neutron-neutron
two-body virtual energy $\epsilon_n$ are smaller than any other energy scale in
the problem, but the scattering between the core and a single neutron is not
fine-tuned, and the Efimov effect does not operate. The theory has one
dimensionless coupling which formally runs to a Landau pole in the ultraviolet.
We show that many properties of the system are universal in the double
fine-tuning limit. The ratio of the mean-square matter radius and charge radius
is found to be $\langle r^2_m \rangle/\langle r^2_c\rangle = A
f(\epsilon_n/B)$, where $A$ is the mass number of the core and $f$ is a
function of the ratio $\epsilon_n/B$ which we find explicitly. In particular,
when $B\gg\epsilon_n$, $\langle r^2_m\rangle/\langle r^2_c\rangle = \frac23 A$.
The shape of the the $E1$ dipole strength function also depends only on the
ratio $\epsilon_n/B$ and is derived in explicit analytic form. We estimate that
for the $^{22}$C nucleus higher-order corrections to our theory are of order
20% or less if the two-neutron separation energy is less than 100 keV and the
$s$-wave scattering length between a neutron and a $^{20}$C nucleus is less
than 2.8 fm.
