AbstractWe introduce so-called consistent posets which are bounded posets with an antitone involution$$'$$′where the lower cones of$$x,x'$$x,x′and of$$y,y'$$y,y′coincide provided thatx, yare different from 0, 1 and, moreover, ifx, yare different from 0, then their lower cone is different from 0, too. We show that these posets can be represented by means of commutative meet-directoids with an antitone involution satisfying certain identities and implications. In the case of a finite distributive or strongly modular consistent poset, this poset can be converted into a residuated structure and hence it can serve as an algebraic semantics of a certain non-classical logic with unsharp conjunction and implication. Finally we show that the Dedekind–MacNeille completion of a consistent poset is a consistent lattice, i.e., a bounded lattice with an antitone involution satisfying the above-mentioned properties.