Phase interaction issues potentially become problematic if the mics are moved apart from each other and the four mics are all in different places, but there shouldn't be phase problems if the mics on each side are positioned coincident with each other. Of course that doesn't stop a lot of people around here from doing just that, but they are powering the mics correctly and either recording all four channels separately, or combining them after the preamp.
The practical problem is electrical. The mics are likely to interact with each other and the input stage of the recorder or preamp in problematic ways if combined at mic level. They may not be powered properly, their performance may suffer dramatically, the load on the input stage of the recorder becomes unpredictable, and the mix between them is likely to not be what you're hoping for. If you want to do this the channels could be combined at line level (after the mic preamp), or the four channels recorded individually and mixed latter which gives you the most flexibility in controlling the mix between the two.
If done correctly, combining a coincident omni and a cardioid mic modifies the overall pickup pattern (polar pattern) to something between that of the two individual mics. If the mics are of equal sensitivity and are combined in equal amounts, you end up with a wide cardioid pattern that is halfway between a cardioid and omni in shape, pointed in the direction of the cardioid. That's exactly how large diaphragm mics with electrically switchable patterns work. They electrically combine two back to back cardioid elements in the mic itself, mixed together in various ratios and with the appropriate polarity to create resulting patterns that range from omni to cardioid to figure 8 or patterns in between.