  The emergent Weyl fermions in condensed matter generally break the Lorentz
invariance resulting in a tilted (type-I) or over-tilted (type-II) energy
dispersion. The tilting energy spectrums can lead to exotic quantum
interference effects in a junction set up. Here, we theoretically investigate
the Josephson current in a Weyl superconductor-Weyl (semi)metal-Weyl
superconductor junction of a time-reversal (TR) broken type-I Weyl semimetal.
We demonstrate that the Cooper pairs of BCS-like pairing acquire a finite
momentum in case of inversion symmetric tilt. Consequently, the system exhibits
tilt induced anomalous current phase relations which are manifested by
supercurrent $0$-$\pi$ transition and Josephson $\phi$ junction. On the
contrary, these effects remain absent in case of inversion breaking tilt and
for FFLO-like pairing in the Weyl superconductor. We further chart out
qualitative differences between the two distinct types of pairings by studying
the critical current dependency on junction length. Our study opens a new
avenue to probe the unconventional superconducting pairings in TR-broken Weyl
semimetals. It is also quite interesting that the tilting in Weyl nodes
naturally leads to anomalous current phase relations in this model without any
magnetic manipulation!
