I was using what they both won when Ron moved to LL but I included his United CL also.
Baring in mind Ronaldo is 2 years older so was already established at United before Messi was.
Well if you want to sound more fair before making statement like "Messi has far more team trophies than Ronaldo", then better do it in a right way.
Domestic Competitions:
Domestic League titles
Messi 7 or 9 (with 2 seasons winning the league being bit part player)
Ronaldo 5
Domestic Cups
Messi 6
Ronaldo 5
Domestic Super Cup
Messi 7
Ronaldo 4
European Competitions
Champions League
Messi 3 or 4 (with 1 trophy won being bit part player)
Ronaldo 5
UEFA Super Cup
Messi 3
Ronaldo 2
World Competitions
Club World Cup
Messi 3
Ronaldo 4
International Competitions
Messi 0 or 2 (with 2 trophy won being u-20 or u-23 competition)
Ronaldo 1
Overall speaking, their team trophies won are not that far apart, in fact, they are more or less similar.
If include everything, such as all one-off super cup matches or charity shields or u-20 competitions etc Messi has 33 (or 36, if including bit part role), Ronaldo has 26.
In terms of major trophies counted (League, CL, CWC, domestic cups, major international competition, excluding all super cups & one-off match), Messi has 19 (or 22, if including bit part role), Ronaldo has 20.
If count only the major prestigious trophies, such as CL, domestic league, international major competitions, excluding CWC, domestic cups & all super cups, Messi has 10 (or 13, if including bit part role), Ronaldo has 11.
Conclusion
Messi has won more team trophies in all levels overall (minor or major), Ronaldo has won more major trophies overall (excluding bit part role)