  Let $(X, Y)$ be a couple of quasi-Banach lattices of measurable functions on
$\mathbb T \times \Omega$ satisfying some additional assumptions. The
K-closedness of a couple of Hardy-type spaces $(X_A, Y_A)$ in $(X, Y)$ and the
stability of the real interpolation $(X_A, Y_A)_{\theta, p} = (X_A + Y_A) \cap
(X, Y)_{\theta, p}$ are shown to be equivalent to each other and to the
BMO-regularity of the associated lattices $\left(L_1, \left(X^r\right)'
Y^r\right)_{\delta, q}$. The inclusion $\left(X^{1 - \theta} Y^\theta\right)_A
\subset \left(X_A, Y_A \right)_{\theta, \infty}$ is also characterized in these
therms. New examples of couples $(X_A, Y_A)$ with this stability are given,
proving that this property is strictly weaker than the BMO-regularity of $(X,
Y)$.
