Is tai chi compatible with the gym?

Gym work typically shortens muscles and promotes patterns of tension and fixity in the body.

Tai chi does the opposite: it allows muscles to lengthen, releases tensions and encourages good skeletal use.
The muscle work that takes place in tai chi is principally geared towards whole-body movement and training the body to rely more upon postural muscles.
 

Body building and weight lifting are fashionable activities today. The emphasis is upon developing external muscles which creates an armouring effect that can eventually distort the bony structure. It is the over developed musculature that actually torque's the bones and discourages them from bearing additional weight. The body attempts to compensate and problems arise.

Running does not necessarily in and of itself improve posture that is already poor and constricted. It often exaggerates problems due to the substitution of inappropriate muscles. The repetitive inappropriate development of the musculature (as in body building or weight lifting) often leads to diminished sensitivity. Stress occurs in the knees and lower back, encouraging injury.

Various sport activities emphasize strength, endurance and speed. Development of muscle control rather than skeletal balance takes precedence. Gaining speed at the expense of mounting tension, is too often the goal. 

(Liz Koch)

The health benefits of tai chi are quickly undermined by weight lifting, exertion and forcing.