  True to their unconventional nature, multi-band alkaline Fe-selenides and,
more recently, the heavy-fermion CeCu$_{2}$Si$_{2}$ have shown signatures of
fully-gapped but sign-changing superconductivity (SC). A two-orbital pairing
state, called $s\tau_{3}$, with non-trivial matrix structure, was proposed as a
candidate able to reconcile the seemingly contradictory properties of these
SC's. Motivated by the non-trivial orbital structure of the proposed
$s\tau_{3}$ state, which has orbital-selective pairing structure, we study
prototypical Josephson junctions where at least one of the leads is in a SC
state of this kind. An analysis of these junctions in the limit of two
degenerate orbitals (bands) and with a simple form of junction hybridization
reveals several remarkable properties. One is the emergence of gapless, purely
electron- and hole-like bound states for $s\tau_{3}-N-s\tau_{3}$ junctions with
arbitrary global phase difference between the leads, and likewise for
$s\tau_{3}-N-I$ junctions. The other is the absence of static Josephson
currents when both leads are SC's. In both of these signatures, $s\tau_{3}$
junctions are dramatically different from conventional Josephson junctions. We
also find that the gapless bound states are protected by an orbital-exchange
symmetry, although the protection is not topological. Junctions which break
this symmetry, such as $s\tau_{3}-N-s$, have gapped Andreev bound states. In
general, the Josephson effect also re-emerges once the degeneracy of the two
orbitals is lifted. We support these conclusions via analytical and numerical
results for the bound states, together with microscopic calculations of the
Josephson current. Our results indicate that junctions involving $s\tau_{3}$
pairing in alkaline Fe-selenides's will generically have bound states with a
small gap together with a greatly suppressed Josephson current.
