The honest core of the word is mutual awkwardness. The romance is a Victorian missionary's free translation that the internet promoted to a definition — which is a good reminder that 'untranslatable' sometimes means 'translated once, by one person, without checking'.
Side by side
Aspect
The famous gloss
The morphology
Suggests
Unspoken romance
Mutual awkwardness
Source
A missionary
Morpheme analysis
In Bridges' dictionary
No
The parts, yes
Checkable
No
Partly
What each one means
Mamihlapinatapai
Mamihlapinatapai is a Yaghan word famous in English for a wordless look between two people who each want the other to act — a definition the linguistic evidence supports far more weakly than the internet suggests.