  Semi-Markov processes generalize Markov processes by adding temporal memory
effects as expressed by a semi-Markov kernel. We recall the path weight for a
semi-Markov trajectory and the fact that thermodynamic consistency in
equilibrium imposes a crucial condition called direction-time independence for
which we present an alternative derivation. We prove a thermodynamic
uncertainty relation that formally resembles the one for a discrete-time Markov
process. The result relates the entropy production of the semi-Markov process
to mean and variance of steady-state currents. We prove a further thermodynamic
uncertainty relation valid for semi-Markov descriptions of coarse-grained
Markov processes that emerge by grouping states together. A violation of this
inequality can be used as an inference tool to conclude that a given
semi-Markov process cannot result from coarse-graining an underlying Markov
one. We illustrate these results with representative examples.
