  We consider a system consisting of two tunnel-coupled two-dimensional
topological insulators proximitized by a top and bottom superconductor with a
phase difference of $\pi$ between them. We show that this system exhibits a
time-reversal invariant second-order topological superconducting phase
characterized by the presence of a Kramers pair of Majorana corner states at
all four corners of a rectangular sample. We furthermore investigate the effect
of a weak time-reversal symmetry breaking perturbation and show that an
in-plane Zeeman field leads to an even richer phase diagram exhibiting two
nonequivalent phases with two Majorana corner states per corner as well as an
intermediate phase with only one Majorana corner state per corner. We derive
our results analytically from continuum models describing our system. In
addition, we also provide independent numerical confirmation of the resulting
phases using discretized lattice representations of the models, which allows us
to demonstrate the robustness of the topological phases and the Majorana corner
states against parameter variations and potential disorder.
