On the international scene he designed costumes for stage productions in Germany, Canada, Caracas, Chile, Brazil, Spain, Uruguay. In the United States he worked for the San Francisco Opera, Washington Opera, Baltimore Opera, Miami Opera, Seattle, Columbus, Ohio, Portland, Los Angeles and Dallas Opera. For the Anna Livia Festival of Dublin he designed Carmen, Hérodiade, Il Trovatore, Gianni Schichi, Il Tabarro and Martha.
Lapiz was the recipient of numerous awards for his work as costume designer: I Due Foscari (1981), Macbeth (1982), Benvenuto Cellini ((1983) and Siegfried (1987). In the United States, in 1989 his costumes for Salome were chosen as the best wardrobe of the year for opera. He received the Drama Critics Award as best costume designer for Turandot, Tosca, Die Fliegender Holländer, Die Walküre, Jenufa, Boris Godunov and Wozzeck.
He designed the costumes for Highlander II, which was filmed in the Teatro Colón in 1991. For the Baltimore he designed the costumes for Tristan and Isolde, Samson and Delilah, Don Carlo, Salome, The Pearl Fishers, Norma, Elektra, Aida, Lucia di Lammermoor and Nabucco.