A process for foam forming tissue or paper webs is disclosed. A foamed suspension of fibers is deposited onto a forming fabric and contacted with a gas flow prior to drying the web. For instance, the web can contact the gas flow prior to dewatering the web. The gas flow can have a volumetric flow rate and/or a velocity sufficient to rearrange the fibers within the web. In one embodiment, for instance, the gas flow can increase the caliper of the web, the stretch properties of the web, and/or the absorbency characteristics of the web. In one embodiment, the gas flow can be pulsed for producing a web with a distinctive pattern.