Yes.So is this a vassal state or a colony/province/whatever of China? There's a huge difference.
So a territory belonging to China, and not a vassal state.
But because of distance it wouldn't be integrated. Many colonies throughout history could be called vassal states, but this would be more like a viceroyalty. It is administrated in the ruler's name, not administrated independently to serve the ruler.
By sparsely-populated north, I meant everything north or northeast of Jalisco, not just the far desert. Also, in this scenario Mexico/New Spain may well include the highly habitable California.Incidentally, I presume you are aware Mexico's sparely populated regions are sparsely populated for a reason - and that rice farming would do extremely poorly there.
And the Chinese farm grain as well as rice. And could adopt corn and beans if they had territory in the area. The natives would keep farming corn and beans, at least.
Mestizos and criollos would support Spain and be horrified at the prospect of being ruled by not only non-Catholics, but from their perspectives, pagans.And why would they be unreliable where as Chinese-descended people would be? The mountains are high and the Emperor is far away, as the saying goes.
I suppose indios (descendants of former Mesoamerican royal families, perhaps) could be reliable rulers of Chinese Mexico. But it really wouldn't be that hard to send settlers, especially if an invasion force had already been sent.