First off, I realize my question is too broad as it depends on the country, but I would be interested to hear mostly about the process in Canada and in western Europe, where it is common to complete a Master's degree before one's PhD.
I am an MSc mathematics student, and I am considering to switch to a different field for my PhD studies. I am thinking about certain directions in Computer Science or Electrical Engineering, which are completely unrelated to my current research.
Within my university such a switch is not uncommon, as the duration of a PhD has quite a soft upper bound, and a student coming from a different background can be accepted and simply required to learn the necessary basics of the new topic. Also, students who have received their MSc from the same university are not required to present official letters of recommendation when applying for a PhD, and the acceptance depends pretty much solely on the new advisor.
However, I imagine such a process would be much more complicated when applying to a different university, especially in a different country, as the department (or the potential advisor) has no way to evaluate the student's potential in the new field (assuming his MSc has nothing to do with the new field), and the student can't present letters of recommendation from researchers in said field.
Are such cases common? Are students coming from a different background at a significant disadvantage when competing with applicants from the same field, and how could one raise their chances of acceptance?
Thanks!