What is a "Mexican Ultimatum"?

Solution 1:

Did you try Google? I searched for "define Mexican Standoff" and the first result, as well as many that followed, said "a situation in which no one can emerge as a clear winner".

Mexican standoff has more in-depth etymology of the expression.

Solution 2:

Well, as far as I know there is no such thing as a 'Mexican ultimatum'.

A Mexican standoff is a fraught situation of 'stalemate'. The classic image is of a ring of gunfighters, all with pistols drawn, with none of them wanting to start shooting. See Anderson Silva's link for more in-depth info.

Solution 3:

While written rather snarky, tvtropes' article gives a bit more insight and criticizes this trope:

A stalemate where everyone has a weapon pointed at someone else. All the threats are equally balanced to ensure a Mutual Disadvantage;

[...]

By the time your enemy realizes they've been shot, the bullet will have exited the back of their head, taking their motor cortex (and thus the capacity to shoot back) with it. However, that wouldn't be nearly as tense and exciting.