  We use the action of the Bockstein homomorphism on the cohomology ring
$H^*(X,\mathbb{Z}_2)$ of a finite-type CW-complex $X$ in order to define the
resonance varieties of $X$ in characteristic 2. Much of the theory is done in
the more general framework of the Maurer-Cartan sets and the resonance
varieties attached to a finite-type commutative differential graded algebra. We
illustrate these concepts with examples mainly drawn from closed manifolds,
where Poincar\'e duality over $\mathbb{Z}_2$ has strong implications on the
nature of the resonance varieties.
