  A $p$-arithmetic subgroup of $\mathrm{SL}_2(\mathbb{Q})$ like the Ihara group
$\Gamma := \mathrm{SL}_2(\mathbb{Z}[1/p])$ acts by M\"obius transformations on
the Poincar\'e upper half plane $\mathcal{H}$ and on Drinfeld's $p$-adic upper
half plane $\mathcal{H}_p :=
\mathbb{P}_1(\mathbb{C}_p)\setminus\mathbb{P}_1(\mathbb{Q}_p)$. The diagonal
action of $\Gamma$ on the product is discrete, and the quotient
$\Gamma\backslash(\mathcal{H}_p\times \mathcal{H})$ can be envisaged as a "mock
Hilbert modular surface". According to a striking prediction of
Nekov\'a$\check{\text{r}}$ and Scholl, the CM points on genuine Hilbert modular
surfaces should give rise to "plectic Heegner points" that encode non-trivial
regulators attached, notably, to elliptic curves of rank two over real
quadratic fields. This article develops the analogy between Hilbert modular
surfaces and their mock counterparts, with the aim of transposing the plectic
philosophy to the mock Hilbert setting, where the analogous plectic invariants
are expected to lie in the alternating square of the Mordell-Weil group of
certain elliptic curves of rank two over $\mathbb{Q}$.
