We investigate the complexity of the evaluation problem for ECRPQ: Conjunctive Regular Path Queries (CRPQ), extended with synchronous relations (a.k.a. regular or automatic). We give a characterization for the evaluation and parameterized evaluation problems of ECRPQ in terms of the underlying structure of queries. As we show, complexity can range between PSpace, NP and polynomial time for the evaluation problem, and between XNL, W[1], and FPT for parameterized evaluation.