The full term is "prova all'italiana". There are numerous references to this online; nearly all of them linked with the German (almost) equivalent "sitzprobe". It is easy to understand the meaning of the German phrase, it is literally a "seated-rehearsal",
where the singers sing with the orchestra, focusing attention on integrating the two groups. It is often the first rehearsal where the orchestra and singers rehearse together.
Crucially this same Wikipedia article continues,
The equivalent Italian term is prova all'italiana.
(Usually I would be sceptical about trusting Wikipedia, but there are numerous other sources of this same information, for instance here and here.)
The Italian term is one of several which use the word "prova" (literally meaning "test") to refer to a rehearsal. For instance, there is also the "prova generale" which refers to a final dress-rehearsal. In German there is the "wandelprobe", where the singers go through the motions of acting their parts while the orchestra plays, and the "generalprobe", again a final dress rehearsal. (See this source.)
Which is all very interesting, but hasn't really answered your central question! Why is this called a "rehearsal-in-the-Italian-style"? This light-hearted article muses on the same question, initially feigning ignorance about the term "sitzprobe", but then acknowledging its meaning, before genuinely being unable to give a reasonable explanation of the term "prova all'italiana", instead resorting to suggesting that it sounds,
as though it should mean to test everything in Italian, but obviously doesn't.
So, why this term? After a couple of hours of internet searches, and even reading some books, I couldn't find you a definitive answer. But it seems that this kind of performance or rehearsal is referred to by a lot of people as a "prova all'italiana", a "rehearsal-in-the-Italian-style", or I prefer the way this guy refers to it,
rehearsal Italian style.