In Wikipedia's article "Complete Boolean algebra, the following example is given:
The Boolean algebra of all Baire sets modulo meager sets in a topological space with a countable base is complete...
This suggests that the Boolean algebra of all Baire sets modulo meager sets in a topological space without a countable base may not be complete.
Is this true?
For instance, consider an uncountable discrete space $X$. The compact $G_\delta$ subsets of $X$ are just the finite sets, so the Baire sets are the sets which are countable or cocountable. The only meager set is $\emptyset$, so the Baire sets modulo meager sets are again the countable or cocountable sets. This algebra is not complete (for instance, if you take a subset $Y\subset X$ which is neither countable nor cocountable, the join of all countable subsets of $Y$ does not exist).