Mixing tire brands and types on the same vehicle isn't the best idea under most circumstances. Sometimes it can be used to make corrections in the vehicle balance or to enhance performance, but most of the time it's a concession best not made. This is especially true with the Honda CR-V, whose unique 4WD system isn't going to appreciate such financial prudence.
In most ways, the Honda CR-V is a taller and more spacious version of its Civic sedan counterpart. Like a station wagon, essentially. In fact, one of the few things that separates the CR-V from the pretentious station wagons that are most crossovers is its Real Time 4WD that is not, contrary to popular belief, an all-wheel-drive system. Real Time 4WD is a kind of auto-locking 4WD system that uses a set of clutches that can engage the rear driveshaft, and vary torque output to that shaft.
Mixing brands of tires on front, rear and all-wheel drive cars isn't necessarily a bad thing, depending upon the result you're trying to achieve. Using a stickier tire on the drive tires -- the rear tires on a RWD car or the front tires on a FWD car -- can give you a bit of extra traction for acceleration or cornering, and help to fine-tune the car's balance. The same is true of AWD cars, but the AWD system is a bit more sensitive to differences in tire diameter and traction coefficients.
The CR-V 4WD system might throw something of a wrench into the works, primarily because it uses a pair of hydraulic pumps to detect front-to-rear slippage rather than individual wheel-speed sensors like many AWD systems. The Real Time system knows only if the rear axle is spinning faster than the front; not a problem but for the CR-V's open differentials that transfer power to the tires with the least traction. When you hit the brakes, the 4WD system disengages so the anti-lock system can work properly.
Using different brands on the front and rear isn't the best idea, since it could mess up the 4WD system's torque transfer. This is especially true when you consider that different manufacturers use different construction methods, which can slightly alter tire diameter when air pressure is applied. But the 4WD's clutches should be able to compensate for minor differences in diameter and traction, much like an AWD system's center diff.
Do not, under any circumstances, use different brands of tires on opposite sides of the same axle. That means you either buy the same brand to replace a blown tire, or replace both tires on the same axle with the same brand and type. The CR-V's open differentials will route power to the tire with the least traction, which means that the better and stickier tire winds up doing nothing in terms of delivering power to the ground. If any one of the tires spins prematurely, the 4WD system will reroute power to the other axle before it's needed. This could overwhelm the other axle's tires, throwing off vehicle stability under acceleration and braking.
