  We study the spin-phonon drag effect in the magnetothermal transport of
spin-1/2 two-leg ladders coupled to lattice degrees of freedom. Using a bond
operator description for the triplon excitations of the spin ladder and
magnetoelastic coupling to acoustic phonons, we employ the time convolutionless
projection operator method to derive expressions for the diagonal and
off-diagonal thermal conductivities of the coupled two-component triplon-phonon
system. We find that for magnetoelastic coupling strengths and diagonal
scattering rates relevant to copper-oxide spin-ladders the drag heat
conductivity can be of similar magnitude as the diagonal triplon heat
conductivity. Moreover, we show that the drag and diagonal conductivities
display very similar overall temperature dependences. Finally, the drag
conductivity is shown to be rather susceptible to external magnetic fields.
