  We define larger variants of the vector spaces one obtains by decategorifying
bordered (sutured) Heegaard Floer invariants of surfaces. We also define
bimodule structures on these larger spaces that are similar to, but more
elaborate than, the bimodule structures that arise from decategorifying the
higher actions in bordered Heegaard Floer theory introduced by Rouquier and the
author. In particular, these new bimodule structures involve actions of both
odd generators $E$ and $F$ of $\mathfrak{gl}(1|1)$, whereas the previous ones
only involved actions of $E$. Over $\mathbb{F}_2$, we show that the new
bimodules satisfy the necessary gluing properties to give a 1+1 open-closed
TQFT valued in graded algebras and bimodules up to isomorphism; in particular,
unlike in previous related work we have a gluing theorem when gluing surfaces
along circles as well as intervals. Over the integers, we show that a similar
construction gives two partially-defined open-closed TQFTs with two different
domains of definition depending on how parities are chosen for the bimodules.
We formulate conjectures relating these open-closed TQFTs with the
$\mathfrak{psl}(1|1)$ Chern--Simons TQFT recently studied by Mikhaylov and
Geer--Young.
