Hi and welcome to the forum.

I think you may be complicating the situation for yourself and for your company with all the questions. Technically, you can't work at all on a tourist visa. There is normally, however, a "business trip" sort of tourist visa that allows you do make the odd business trip to do work for your employer over a period that does not exceed either 3 or 6 months at a time. On that sort of visa, there is no tax issue except with your home country (i.e. your country of residence, not necessarily your nationality) over traveling expenses. It might be possible to get a multiple entry business visa that would allow you to make trips lasting up to 3 - 6 months at a time over one or several years. However...

If you become RESIDENT in Canada, then you are subject to Canadian income tax on your earnings in Canada - no matter where your company is located (or even where they are paying you or in what currency). What it takes to become tax resident gets complicated - but for most countries, if you are spending 183 days a year in the country there is a presumption that you probably live there and have your primary "center of interest" there (even if you are moving between different hotels every few weeks). And that's when the fun starts...

How is your company proposing that you do these consulting gigs? If they are going to want you in Canada (or the US) for more than 183 in a given year, they should be the ones applying for a visa for you that will allow you to work as they want you to.
Cheers,
Bev