  Let $F$ be a totally real field and $p$ a rational prime unramified in $F$.
For each subset $P$ of primes of $F$ above $p$, there is the notion of
partially classical Hilbert modular forms, where $P=\varnothing$ recovers the
overconvergent forms and $P$ being the full set of primes above $p$ recovers
classical forms. Given $P$, we $p$-adically interpolate the classical modular
sheaves to construct families of partially classical Hilbert modular forms with
weights varying in appropriate weight spaces. We then construct the
corresponding eigenvariety, which recovers the eigenvariety constructed by
Andreatta--Iovita--Pilloni when $P=\varnothing$. By contrast, it is unclear
whether each partially classical form can be put into a partially classical
family.
