We present a family of paraconsistent counterparts of the constructive modal logic \(\textsf{CK}\) . These logics aim to formalise reasoning about contradictory but non-trivial propositional attitudes like beliefs or obligations. We define their Kripke-style semantics based on intuitionistic frames with two valuations which provide independent support for truth and falsity; they are connected by strong negation as defined in Nelson’s logic. A family of systems is obtained depending on whether both modal operators are defined using the same or by different accessibility relations for their positive and negative support. We propose Hilbert-style axiomatisations for all logics determined by this semantic framework. We also propose a family of modular cut-free sequent calculi that we use to establish decidability.