  Rigid meromorphic cocycles were introduced by Darmon and Vonk as a
conjectural $p$-adic extension of the theory of singular moduli to real
quadratic base fields. They are certain cohomology classes of
$\mathrm{SL}_2(\mathbb{Z}[1/p])$ which can be evaluated at real quadratic
irrationalities and the values thus obtained are conjectured to lie in
algebraic extensions of the base field. In this article we present a similar
construction of cohomology casses in which $\mathrm{SL}_2(\mathbb{Z}[1/p])$ is
replaced by an order in an indefinite quaternion algebra over a totally real
number field $F$. These quaternionic cohomology classes can be evaluated at
elements in almost totally complex extensions $K$ of $F$, and we conjecture
that the corresponding values lie in algebraic extensions of $K$. We also
report on extensive numerical evidence for this algebraicity conjecture.
