Can a relationship that began while one partner was still legally married to someone else still be sponsored?

Yes, this can still be sponsored, but it requires care. Once the earlier marriage has properly ended, for example through a legally recognized divorce, and the current relationship can be shown to be genuine when the sponsorship application is made, the fact that it began while one partner was still legally married to someone else does not automatically disqualify a later, real relationship from being sponsored.

Where this gets more complicated is that the overlap in timing is exactly the kind of detail that invites closer scrutiny of the relationship's whole history. IRCC may want to understand when the new relationship actually started, how it developed alongside the ending of the earlier marriage, and whether the timeline as a whole supports a genuine relationship rather than raising questions about it. The single most important thing here is complete, honest disclosure of the full timeline, including dates, the status of the earlier marriage at each point, and how the new relationship progressed, rather than downplaying the overlap. Attempting to hide or simplify an awkward timeline tends to create bigger problems than disclosing it honestly ever would. Because these situations are genuinely fact-sensitive and the framing matters, it is worth working with an immigration lawyer to prepare a clear, complete, honest account before applying.