Having just worked through my initial color management migraine, here’s a generic explanation that might help you put whatever RT is doing, in context.
Fundamental concept: ICC-based color management requires from the start for the image to have an associated color profile that is used to transform the image at various places for working, display, and output (well, display is really a form of output…). This transformation is simplistically thought of as, e.g., working->output, but is really working->CIEXYZ->output. Profiles thusly don’t have to know about each other, they just know how to get back and forth from/to CIEXYZ, the “mother profile” you can read about elsewhere.
So, your question relates to the difference between ‘display’ and ‘output’. I’ll assume you’ve been munging your image in whatever software using a decent working profile. You probably know that displaying your image with that profile ‘straight to the lcd’ looks bad, because your display’s colorspace is woefully inadequate to display the internal richness of the working image. So, prior to pushing the pixels to the lcd, your software transforms the working image to a display image using the calibrated display profile. Looks nice.
Now, you want to save your image for passing to friends and family, but you don’t know jack about their displays. So, you save/export your working image to a JPEG, because you know most folks can look at those, and you transform the working image to a ‘least-common-denominator’ output colorspace, e.g., sRGB. Really, if everyone in the whole world were using color-managed software that displayed 16-bit TIFF or PNG you wouldn’t have to do this transform, you could just save the working image with it’s working profile and your cousin’s software would do the working->CIEXYZ->display transform. But alas, we’re not there yet, maybe when we have self-driving cars we’ll have universal color management… .
sRGB actually does something else important besides following the image around with a corresponding color profile; transforming to the limited sRGB colorspace makes the image ‘default-displayable’ in non-color-managed softwares, (hopefully) approximating the colors you want folk to see. Even among color-managed viewers, not all is equivalent; some only completely recognize V2 ICC profiles and ignore V4 profiles or worse. You can see this with the old Windows Photos Viewer (not the Photos app) - save your working image to JPEG with a V4 sRGB profile, and this viewer shows it different than your RT display. Save it with the V2 sRGB profile, and now it looks the same. Download @Elle’s profiles and you can easily demonstrate this.
@Jossie, you probably know a lot of the above, so I apologize if the above re-treads your understanding. But it seemed your question relates to the difference in handling internal display vs export to a file, so I thought I’d walk those two paths in a single post. Sometimes, methinks the dialogue in forum posts doesn’t fully connect the dots (well, that might be my old-age dementia…)
Speculating, I think you might be experiencing the V2/V4 thing…