A tissue sheet is made using a modified wet pressing machine in a modified wet pressing process employing an integrally sealed air press. After initial formation and conventional vacuum dewatering, the wet web is conformed to the surface contour of a relatively coarse fabric to give the web a textured surface. By creating a pressure differential across the web of at least 30 inches of mercury and an air stream through the web of at least 500 SCFM/in2, the air press noncompressively dewaters the wet web to a consistency of about 30 to about 40 percent prior to a heated drying cylinder. The web is dried to substantially preserve its three-dimensional, throughdried-like texture. The process provides a web having an exceptionally high degree of bulk and absorbency not expected in wet-pressed products.