  An abstract property (H) is the key to a complete a priori error analysis in
the (discrete) energy norm for several nonstandard finite element methods in
the recent work [Lowest-order equivalent nonstandard finite element methods for
biharmonic plates, Carstensen and Nataraj, M2AN, 2022]. This paper investigates
the impact of (H) to the a posteriori error analysis and establishes known and
novel explicit residual-based a posteriori error estimates. The abstract
framework applies to Morley, two versions of discontinuous Galerkin, $C^0$
interior penalty, as well as weakly over-penalized symmetric interior penalty
schemes for the biharmonic equation with a general source term in
$H^{-2}(\Omega)$.
