Romanish
The term Romanische or some close variant has historically been used by the Germanic peoples as an identifier for the Roman diaspora, those people still speaking the Roman/Latin language on the borders of the former Roman Empire. In the German language, the term Rumänisch is still used to refer to Romanians. The word Rumantsch is used to refer to the Roman diaspora beyond the Alps, namely in eastern Switzerland. The word Romanish is an attempt to define this concept, Roman diaspora, in English.
This site then, outlines research regarding the Roman diaspora, especially those given some render of the Germanic identifier "Walhaz". This is not a resource that argues for or against the "Romanity" of the "Eastern" Roman Empire, although it is evident that the word "Byzantine" is politically loaded and pejorative, and the terms Romaion Empire or even Romaic Empire better define the post-Justinian Roman Empire administered from Constantinople, rather than continuing the polarizing dichotomy of "east" and "west", or playing into concepts of "racial hierarchy" .
On the word "Walhaz"
The word "Walhaz" is a reconstructed proto-Germanic word, that was used to refer to Romans, Latin speakers, by the Germanic peoples. This was their pronunciation of the Celtic tribe "Volcae", recorded by the Greeks and Romans as living at either end of the Alps in antiquity.
As the Roman Empire expanded, the Volcae became Roman through military conscription, voluntary service, trade advantages, etc, and started speaking the Roman language of Latin. They are the medium through which the Germanic peoples came to understand the Roman identity, language, customs, and so forth.
The Germanic pronunciation is still evident in a range of peoples who live at the borders of the Roman Empire, and who spoke the Latin language, or still do.
The word survives through the Welsh in the British Isles who grew out of the fusion of the Celtic and Latin speaking population after the Roman Empire withdrew from Britain in 410 CE. The Walloons of northeastern France, Belgium and the Netherlands carry the root word as well, and grew out of the same phenomenon, a Latin speaking community after the Roman Empire withdrew from the region. The Welsch (referred to as Romansh in English, and now known as Rumantsch in German) of Switzerland carry the same word, and descend from the same phenomenon. The Wallachs of Romania carry the same word, and again descend from the same phenomenon. The Vlachs of (namely) Greece are yet another example.
Worth noting that the Polish word for an Italian to this day is Wloch.
Most of these groups call themselves Romans to this day, including the Romanians, Romansh, Aromanians, etc. Even phonetically, if you listen to the pronunciation of the ethnonym, the term they call themselves, it is plainly obvious they are identifying themselves as Romans. They are called by different terms in English, just as "Germans" call themselves "Deutsch", or "Greeks" call themselves "Hellenes".
On the value of definition
Most of these exonyms were equated or adapted in a pejorative manner at some point, for instance, the English word "welch" meant "to steal, to go back on your word", the Greek word "vlakas", already in existence before the Romans colonized the eastern Adriatic, means "soft, foolish, stupid", while the proto-Germanic word "walaz", meant "corpse". These words were equated with the Romance language speaking neighboring groups, much like the word "gyp" or "jew" came to mean "to offer compensation lower than the value of the material good or service provided". For that matter, the use of the term "Romani" to describe the people group tracing to Northern India known as "gypsies", has been used mostly since 1976 after a U.N. affiliated European council determined the word "gypsy" should be replaced. Historically, the term "Romani" meant "Roman citizens", "Romanus" being the Latin singular for the same, while "Roma" is the name for the city of Rome, in the Latin language. How fantastic to call the people who were referred to as a slang term for "Egyptians", "Romans" instead, centuries and centuries after the Roman Empire fell! These people called themselves an unwritten term that sounds phonetically a lot more like "Doma" and "Domari" than "Roma" and "Romani", but if you want to undermine what it means to be "Roman", and create confusion, you can equate groups of people with different ideas that serve to otherize them, and control the narrative in your society. This has been done over and over and over.
Why the term "Roman" matters
Since long before the founding of the city of Rome, the term "rome" in Greek,"ρωμη", has meant and still does mean "strength, vigor, intention". You can see this concept in the term "Romantic", for someone who chooses a person or an idea and takes steps to actualize making that person their spouse, or that idea into reality. Sure, the Roman Empire fell, it is no more. But dear reader, you are reading in the Latin alphabet. The language you are reading in, English, is 60% Latin vocabulary. Jack White stated in the song "Black Bat Licorice", "The Romans are gone, they changed their name because they lost". Well, not all of them lost, and not all of them changed their name either (Jack White actually changed his name, when he married, taking his wife's last name. His family name is Gillis). Romanians still call themselves Romans, Aromanians call themselves Romans, the word Roman can be heard any time a member of the Romansh people refers to their ethnic group or language. The Romans are still here.
These groups and others will be explored on this resource, where you can learn about the Romans and what they have been up to since Rome fell, and the Roman Empire was conquered.
Special thanks to Thede Kahl (whose wife is Aromanian), chairman of the research and presentation project V.L.A.C.H., Vanishing Language and Cultural Heritage (https://www.oeaw.ac.at/vlach/) which seeks to preserve the rights and languages and cultural heritage of minority groups in Europe and the world over.