Can days working abroad for a Canadian business count toward my permanent resident residency obligation?

In some circumstances, yes. Time spent outside Canada working full-time for a Canadian business, or for the federal public service, under a qualifying employment arrangement can count toward the residency obligation, rather than simply counting as time away from Canada. This is one of the recognized exceptions to the general requirement of physical presence in Canada.

The key word is qualifying: the arrangement has to meet specific conditions about the nature of the employment and the person's ongoing connection to the Canadian business, not simply being employed by a company that happens to be Canadian or that has some Canadian operations. Working for a foreign subsidiary, or in a role with only a loose or indirect connection to the Canadian business itself, is unlikely to meet this exception on its own.

Because this exception is fact-specific and easy to assume applies more broadly than it actually does, anyone relying on foreign work experience to meet the residency obligation should have the specific employment arrangement assessed against the current qualifying conditions before treating that time as automatically counted.