  We present a numerical study of the field-angle resolved oscillations of the
thermal conductivity and specific heat under rotated magnetic field in the
$A_y$Fe$_{2-x}$Se$_2$ [$A$=K,Rb,Cs,(Tl,K)] superconductors, using realistic
two-band Fermi surface parameterization. Our key finding is that even for
isotropic pairing on an anisotropic Fermi surface, the thermodynamic quantities
exhibit substantial oscillatory behavior in the superconducting state, even
much below the upper critical field. Furthermore, in multiband systems the
competition of anisotropies between two Fermi surfaces can cause a double sign
reversal of oscillations as a function of temperature, irrespective of gap
anisotropy. Our findings put severe constraints on simple interpretations of
field-angle resolved measurements widely used to identify the angular structure
of the superconducting gap.
