He was christened Frans (and if Wikitree could make this name fields render sensibly throughout the site, we'd call it his proper name). He appears in records during the New Netherland period as François. When New Netherland became New York, he became Francis. Unless a translation was made of his name in the recording of his will, then he named himself and signed it Francis Rombouts.
Rombout, Rumbout, Rumbouts are all alternate spellings which appear in records, most notably in reference to the "Rombout" Patent. Descendants who were given his name had it spelled Rombout.
As stated as his marriage to Aeltje, Frans was from Hasselt,  located in Liège at the time. Hasselt is now a part of Belgium.
Haßelt, Leodi c. 1570
1665 May 31 Francoijs Rombouts, j.m. Van Hasselt, in't land Van Luijck, Aeltje Wessels, j.d. Van't Tolhuijs bij Schenckenschaus. 
1666 Aug 12 Johannes, Francois Rombout, Aeltje Wessels. Wit.: Warner Wessels, Marritie Wessels. 
1675 Aug 08 Francois Rombouts, Wedr. Van Aeltje Wessels, en Anna Elisabeth Maschop, Wede. Van Warnart Wessels, beijde woonende alhier. 
1683 Sep 08 Francois Rombout, laest Wedr. Van Anna Elisabeth Masschot, en Helena Teller, laest Wede. Van Jan Hendrickszen Van Balen, beijde woonende alhier. Getrouwt den 24 Oct. 
↑ Thomas Grier Evans. "Baptisms from 1639 to 1730 in the Reformed Dutch Church, New York." In Collections of the New-York Genealogical and Biographical Society. Vol. II. New York: Printed for the Society, 1890.
↑ 4.04.1 Samuel S. Purple. "Marriages from 1639 to 1801 in the Reformed Dutch Church, New York" In Collections of the New-York Genealogical and Biographical Society. Vol. I. New York: Printed for the Society, 1890.