  We identify a class of composite membranes: fluid bilayers coupled to an
elastic meshwork, that are such that the meshwork's energy is a function
$F_\mathrm{el}[A_\xi]$ \textit{not} of the real microscopic membrane area $A$,
but of a \textit{smoothed} membrane's area $A_\xi$, which corresponds to the
area of the membrane coarse-grained at the mesh size $\xi$. We show that the
meshwork modifies the membrane tension $\sigma$ both below and above the scale
$\xi$, inducing a tension-jump $\Delta\sigma=dF_\mathrm{el}/dA_\xi$. The
predictions of our model account for the fluctuation spectrum of red blood
cells membranes coupled to their cytoskeleton. Our results indicate that the
cytoskeleton might be under extensional stress, which would provide a means to
regulate available membrane area. We also predict an observable tension jump
for membranes decorated with polymer "brushes".

Soft Condensed Matter