  A relativistic analytical theory of strong field ionization applicable across
the regimes of the deep-tunneling up to the over-barrier ionization (OTBI) is
developed, incorporating the effects of the polarization of the atomic bound
state and the Stark-shift in an ultrastrong laser field. The theory, in
particular, addresses the order of magnitude discrepancy of the ionization
yield in the relativistic regime calculated via the numerical solution of the
Klein-Gordon equation [B. Hafizi \textit{et al}., Phys. Rev. Lett. 118, 133201
(2017)] with respect to the state-of-the-art quasiclassical theory of
Perelomov-Popov-Terent'ev (PPT) for strong field ionization. The developed
theory employs a Keldysh-like approach describing the ionization as an
adiabatic quantum jump from the bound state to the continuum at a specific
transition time, where the improved performance is achieved by accounting for
the bound state distortion in the laser field. In the nonrelativistic limit,
our theory reproduces the well-known fit to the numerical calculations for the
OTBI rate via the Tong-Lin factor. Realistic conditions for an experimental
confirmation of the prediction of the present relativistic model versus
PPT-theory are also presented.
