If a rational relation is total — every input string has at least one output — then it contains an unambiguous rational relation (Lemma B.2.5 of Transducers). After eliminating ε-transitions, the unambiguous automaton follows the run of the original automaton that is lexicographically minimal for a chosen linear order on the states: its states are pairs $P \ni p$ of the set $P$ of states that can still reach acceptance, chosen deterministically from right to left, and the minimal state $p \in P$ reachable so far, chosen deterministically from left to right; the output of a transition is one chosen output of the original automaton between its two states on its letter.