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Unstable regimes occurring for multiphase flow in vertical risers have successfully been stabilized using conventional linear control techniques. However, these control systems rely on downhole measurements which are at best unreliable, if at all available. In this paper, we design a nonlinear observer for the states of the multiphase flow that relies on topside measurements only, and apply it to estimate downhole pressure for feedback control. A key feature of the design is that it exploits the structure of the model to obtain robustness with respect to the internal flows in the system. Combining the nonlinear observer with conventional PI control of the downhole pressure, we demonstrate in laboratory experiments the potential for increasing production from gas-lift wells by stabilizing the multiphase flow.