We're eyeballing a very small scan on the web where many variables can also come into play.
That said, it does look like gross, muggy underexposure to me, and an uncommon result from RDPIII exposed competently in MF or LF. Did you take notes of your exposure to refer back to?
If this was a long exposure (evening?), was allowance made for reciprocity?
A well-exposed, evenly-toned slide does not resemble the sample you have posted, irrespective of the viewing illumination. Also, it would not need, nor benefit from, lots of concentrated bright light, certainly not a 4x5 sheet. In fact that can make critical assessment of highlights, mid-tones and shadows problematic and misleading. If there is little detail visible in great swathes of black shadow, the exposure is grossly out.
Provia and Velvia are designed for ideal exposure in diffuse light, where shadows are softened/filled. In such conditions, highlights can also be preserved. Thus, if shadows are close to or solid black in either RVP50 or RDPIII (the former has a bit more forgiving leeway with contrast but it can still be tipped over without care), the films are being compromised; pulling at development only further compromises the film. It's best to nail the exposure in-camera than resort to a lower-rung quick fix after the event. The erroneous exposure of these two films in conditions known to hobble their capacity to handle both highlights and shadow is a maddenly recurring theme.
At this time though, without a full brief of background information about exposure and conditions, a definite answer cannot be provided. It is suggested you shoot your scene again, recording notes of what you are doing as a very valuable way of referring back to any problems that are immediately obvious. It's not just about development of E6, but critically, how individual sheets and whole rolls were exposed, shot-by-shot.