We report observations of tunneling anisotropic magnetoresitance (TAMR) in vertical tunnel devices with a ferromagnetic multilayer-(Co/Pt) electrode and a non-magnetic Pt counter-electrode separated by an AlOx barrier. In stacks with the ferromagnetic electrode terminated by a Co film the TAMR magnitude saturates at 0.15% beyond which it shows only weak dependence on the magnetic field strength, bias voltage, and temperature. For ferromagnetic electrodes terminated by two monolayers of Pt we observe order(s) of magnitude enhancement of the TAMR and a strong dependence on field, temperature and bias. Discussion of experiments is based on relativistic ab initio calculations of magnetization orientation dependent densities of states of Co and Co/Pt model systems.PACS numbers: 85.75. Mm,75.45.+j,75.50.Cc Anisotropic magnetoresistance (AMR) sensors replaced in the early 1990s classical magneto-inductive coils in hard-drive readheads launching the era of spintronics. Their utility has, however, remained limited partly because the response of these ferromagnetic resistors to changes in magnetization orientation originates from generically subtle spin-orbit (SO) interaction effects [1]. Currently widely used giant magnetoresistance [2] and tunneling magnetoresistance (TMR) [3] elements comprising (at least) two magnetically decoupled ferromagnetic layers provided a remarkably elegant way of tying the magnetoresistance response directly to the ferromagnetic exchange splitting of the carrier bands without involving SO-coupling. Large magnetoresistances in these devices are, nevertheless, obtained at the expense of a significantly increased structure complexity, necessary to guarantee independent and different magnetization switching characteristics and spin-coherence of transport between the ferromagnetic layers.Studies of AMR effects [4,5,6,7] in ferromagnetic semiconductor tunneling devices showed that AMR response can in principle be huge and richer than TMR, with the magnitude and sign of the magnetoresistance dependent on the magnetic field orientation and electric fields. Subsequent theoretical work predicted [8] that the tunneling AMR (TAMR) effect is generic in ferromagnets with SO-coupling, including the high Curie temperature transition metal systems. A detailed investigation of the TAMR is therefore motivated both by its intricate relativistic quantum transport nature and by its potential in more versatile alternatives to current TMR devices which will not require two independently controlled ferromagnetic electrodes and spin-coherent tunneling.Experimental demonstration of the TAMR in a tunnel junction with a ferromagnetic metal electrode has recently been reported [9] in an epitaxial Fe/GaAs/Au stack. The observed TAMR in this structure is relatively small, bellow 0.5%, consistent with the weak SO-coupling in Fe. In this paper we present a study of vertical tunnel devices in which the ferromagnetic electrode comprises alternating Co and Pt films. We build upon the extensive literature [10,11,12,13,14,15] o...