This is interesting. I had vaguely heard of Coriolis flow meters from my time in the oil industry, but never looked into the principle of operation. So thanks for posting this query. I have learnt something.
I think I've got it. It's the inertia of the flowing fluid inside the tube. The arrows show what takes place during the "upward" phase of oscillation of the tube. Fluid entering from the left resists being made to flow slightly upward due to its inertia, creating a downward force on that part of the tube. Conversely, the fluid leaving, back to the left again, has by now been made to flow slightly upward and therefore resists being made to flow horizontally once more, creating an upward force on that part of the tube. So, seen end-on from the right, the tube will have a slight twist as shown. When the tube is in the "downward" phase of its oscillation, the converse happens. When it is in the centre, there is no twist. So, again as seen end-on from the right, there will be a rocking or twisting motion superimposed on the up-down oscillation. In effect, on the side the fluid leaves, the phase is slightly advanced relative to the oscillation with no flow, while on the side the fluid enters the phase is slightly retarded. The downstream side leads the upstream side.
The magnitude of the force will depend on the rate of flow of mass, because when it flows faster more mass has to be made to change direction in unit time, i.e. the rate of change of momentum, d(mv)/dt is greater.
d(mv)/dt = ma = F.
And then of course there a load of fancy stuff about detecting this distortion of the resonant frequency via phase shift etc.