Is it possible to do a simple 2d code with upwinding for the convective fluxes? Everything seem to work fine in 1D, but adding the second dimension is quite a mess ...

yes it is possible...its done all the time. its even a bigger mess when u go 4 3D...

The fact is that while in 1d the flux splitting is quite straightforward, in 2d it not so ... especially when the flow is aligned in one grid direction (roughly). Any practical suggestions?

What I am guessing is your scheme is upwind when the mach number is greater then 1, right ?

So I guess the problem you are facing is that when you have 2D, the flow along x direction maybe supersonic and along y its subsonic or vice versa....

So based on my above assumptions on what the problem is, this is my suggestion. If you are traversing along x directions or I lines, check if the dot product of the velocity vector and normal to the edge (along J line or y dir) has magnitude greater then a (speed of sound), and use upwinding accordingly. Same thing holds when traversing along y direction.

I hope this is what you wanted to know, otherwise post more details about the method you are using, like FD or FV, and the scheme etc.