  Given a Hilbert modular form for a totally real field $F$, and a prime $p$
split completely in $F$, the $f$-eigenspace in $p$-adic de Rham cohomology of
the Hilbert modular variety has a family of partial filtrations and partial
Frobenius maps, indexed by the primes of $F$ above $p$. The general plectic
conjectures of Nekovar and Scholl suggest a "plectic comparison isomorphism"
comparing these structures to etale cohomology. We prove this conjecture in the
case $[F : \mathbf{Q}] = 2$ under some mild assumptions; and for general $F$ we
prove a weaker statement which is strong evidence for the conjecture, showing
that plectic Hodge filtration has a canonical splitting given by intersecting
with simultaneous eigenspaces for the partial Frobenii.
  (In memory of Jan Nekovar)
