  Finding a maximum cardinality common independent set in two matroids (also
known as \textsc{Matroid Intersection}) is a classical combinatorial
optimization problem, which generalizes several well-known problems, such as
finding a maximum bipartite matching, a maximum colorful forest, and an
arborescence in directed graphs. Enumerating all maximal common independent
sets in two (or more) matroids is a classical enumeration problem. In this
paper, we address an ``intersection'' of these problems: Given two matroids and
a threshold $\tau$, the goal is to enumerate all maximal common independent
sets in the matroids with cardinality at least $\tau$. We show that this
problem can be solved in polynomial delay and polynomial space. Moreover, our
technique can be extended to a more general problem, which is relevant to
Matroid Matching. We give a polynomial-delay and polynomial-space algorithm for
enumerating all maximal ``matchings'' with cardinality at least $\tau$,
assuming that the optimization counterpart is ``tractable'' in a certain sense.
This extension allows us to enumerate small minimal connected vertex covers in
subcubic graphs. We also discuss a framework to convert enumeration with
cardinality constraints into ranked enumeration.
