  We study two microscopic models of topological insulators in contact with an
$s$-wave superconductor. In the first model the superconductor and the
topological insulator are tunnel coupled via a layer of scalar and of randomly
oriented spin impurities. Here, we require that spin-flip tunneling dominates
over spin-conserving one. In the second model the tunnel coupling is realized
by an array of single-level quantum dots with randomly oriented spins. It is
shown that the tunnel region forms a $\pi$-junction where the effective order
parameter changes sign. Interestingly, due to the random spin orientation the
effective descriptions of both models exhibit time-reversal symmetry. We then
discuss how the proposed $\pi$-junctions support topological superconductivity
without magnetic fields and can be used to generate and manipulate Kramers
pairs of Majorana fermions by gates.
