Yes, that is always the case, but most of the time, we do not do model comparison. Also, for easy models we can use a prior distribution that excludes the indeterminacy region (like a uniform distribution from 1 to 5 for the inflation feedback coefficient, where indeterminacy happens for values below 1).
If you read the discussion to An/Schorfheide (2006), the discussants are worried about the implicit prior truncation and An/Schorfheide respond that their implicit prior only truncates 3% of the mass and argue that this does not affect their conclusions.
The problem is not having different regimes, but rather having a prior distribution that includes two regimes, but then estimating the marginal data density for the regimes separately. Because in that case, you try to compare to “models” that now have an implicit prior that does not integrate to 1 (because the mass is split across the two regimes).