Was excited to be able to use the MBC in low latency live mode but it seems to be buggy. Please try this in 6.03:
Open empty project
Add instrument track and pick Monologue
Add the Multiband Compressor and pick preset Bass Drum 1
Play a C4 - sounds clear but with latency
Turn on the “Live” button down on the right on the MBC
Play the C4 again - the sine wave is now distorted in a kind of ring modulated way.
Windows XP sp. 3 Core 2 Quad 2.66 GHz, 2 GB RAM.
/K
To fix this bug the MBC was changed and after this it would need more CPU and to process the signal coming in a more high quality way it needed some time for the high-quality processing. Most of the HQ MBC’s on the market introduce such latency by the way. High CPU consumption and latency is common for Multiband Compressors.
After this fix has been introduced there where complaints and a thread got stirred up in which people wanted the old MBC back and would actually prefer the old MBC with less CPU consumption and latency on the cost of artifacts It would not be accepted that the MBC has changed. For these people the MBC live mode has been introduced. In this mode the MBC will not take so much time and CPU to calculate the signal in HQ mode, but it will actually introduce artifacts under certain conditions.
I am not hearing what you are hearing but I assume that you are hearing the artifacts that the old MBC introduced and that the new one introduces in the live mode. Use the HQ mode and the “plugin delay compensation” button during recording instead. 
Gr,
JHP