  We present a new perturbative expansion for pionless effective field theory
with Coulomb interactions in which at leading order the spin-singlet
nucleon-nucleon channels are taken in the unitarity limit. Presenting results
up to next-to-leading order for the Phillips line and the neutron-deuteron
doublet-channel phase shift, we find that a perturbative expansion in the
inverse 1S0 scattering lengths converges rapidly. Using a new systematic
treatment of the proton-proton sector that isolates the divergence due to
one-photon exchange, we renormalize the corresponding contribution to the
3H-3He binding energy splitting and demonstrate that the Coulomb force in
pionless EFT is a completely perturbative effect in the trinucleon bound-state
regime. In our new expansion, the leading order is exactly isospin-symmetric.
At next-to-leading order, we include isospin breaking via the Coulomb force and
two-body scattering lengths, and find for the energy splitting
(E_B(3He)-E_B(3H))^NLO = (-0.86 +/- 0.17) MeV.
