  We develop a relativistic Coulomb-corrected strong field approximation (SFA)
for the investigation of spin effects at above-threshold ionization in
relativistically strong laser fields with highly charged hydrogen-like ions.
The Coulomb-corrected SFA is based on the relativistic eikonal-Volkov wave
function describing the ionized electron laser-driven continuum dynamics
disturbed by the Coulomb field of the ionic core. The SFA in different
partitions of the total Hamiltonian is considered. The formalism is applied for
direct ionization of a hydrogen-like system in a strong linearly polarized
laser field. The differential and total ionization rates are calculated
analytically. The relativistic analogue of the Perelomov-Popov-Terent'ev
ionization rate is retrieved within the SFA technique. The physical relevance
of the SFA in different partitions is discussed.
