  In this reply, we resolve the apparent discrepancy raised in the "Comment on
Inferring broken detailed balance in the absence of observable currents"
[arXiv:2112.08978v1]. We stress that the non-instantaneous transition paths
originate from the choice of a decimation process non-local in time, in
contrast to the decimation process described in our original work, which
commutes with time-reversal. Therefore, the example using such a non-local
decimation procedure, which indeed gives rise to spurious time-irreversibility,
does not invalidate our main result, which was rigorously derived and proven
mathematically. We hope this reply highlights the subtlety of the different
choices for decimating time series data which is pivotal for a correct
understanding of the thermodynamics of small systems.
