  Geometry can be used to explain many properties commonly observed in real
networks. It is therefore often assumed that real networks, especially those
with high average local clustering, live in an underlying hidden geometric
space. However, it has been shown that finite size effects can also induce
substantial clustering, even when the coupling to this space is weak or non
existent. In this paper, we study the weakly geometric regime, where clustering
is absent in the thermodynamic limit but present in finite systems. Extending
Mercator, a network embedding tool based on the Popularity$\times$Similarity
$\mathbb{S}^1$/$\mathbb{H}^2$ static geometric network model, we show that,
even when the coupling to the geometric space is weak, geometric information
can be recovered from the connectivity alone. The fact that several real
networks are best described in this quasi-geometric regime suggests that the
transition between non-geometric and geometric networks is not a sharp one.
