In this work, dedicated to Shafi Goldwasser, we consider a relaxation of the notion of pseudodeterministic algorithms, which was put forward by Gat and Goldwasser (ECCC, TR11-136, 2011). Pseudodeterministic algorithms are randomized algorithms that solve search problems by almost always providing the same canonical solution (per each input). Multi-pseudodeterministic algorithms relax the former notion by allowing the algorithms to output one of a bounded number of canonical solutions (per each input). We show that efficient multi-pseudodeterministic algorithms can solve natural problems that are not solveable by efficient pseudodeterministic algorithms, present a composition theorem regarding multi-pseudodeterministic algorithms, and relate them to other known notions.