The consistency criterion requires that any candidate who wins both of two separate sets of voters must also win the combined electorate. In this definition, "winning" also includes being tied for first place: if a candidate X ties with some other candidate in the first electorate, and wins outright in the second, X must win outright in the combined electorate.
Although it is not a logical requirement, generally speaking systems which satisfy the consistency criterion also satisfy the participation criterion. The reverse implication is also common, though slightly less so.
A voting system is consistent if, when the electorate is divided arbitrarily into two parts and separate elections in each part result in the same alternative being selected, an election of the entire electorate also selects that alternative. If a voting system is not consistent then it may be manipulated through the establishment of strategically configured election districts.
A scale invariant preferential voting method is consistent iff its win regions are convex.
A strict preferential voting method is "consistent if and only if it is a scoring function", i.e. a weighted positional method or a combination of these where one or more weighted positional methods are used in sequence to break the ties of another. If the preferential voting method admits weak preference orders (rankings with equal-rank or truncation), it must reduce to a scoring function when no voters make use of equal-rank or truncation.
In addition, Approval voting and Score voting are consistent. If X is the winner in the first district, then that means that X's score is greater than or equal to any other Y, and the same for the second district. Then summing the districts' scores can not make any Y's sum exceed X's sum.
It has been argued that failure of the consistency criterion can make a voting method more susceptible to gerrymandering, since gerrymandering is essentially about splitting voters into different groups in order to get a certain more-preferred set of election outcomes from each group. See https://www.reddit.com/r/EndFPTP/comments/fhkksz/what_do_you_all_think_about_the_participation/fkbwvz5/.