How can one remain anonymous on secure company Wi-Fi in which you can only login in with company issued password?
I am writing this under the assumption that the only thing you don't control is the WiFi network available to you.
1) Your company doesn't passively monitor their network - they don't care about VPN tunnels, etc. You're worried about a bored Systems Administrator judging you for browsing My Little Pony fan sites during your break. You can't stand to see the judgy look in their eyes when you run into them at the water cooler.
2) Your company doesn't care if you use their network or not.
3) Your company doesn't care what you do on your device.
4) You don't care who snoops on you, as long as it's not your employers.
These are all big assumptions, but you haven't supplied any information to the contrary. If these assumptions are wrong, some or all of this answer will not be appropriate.
You have a few options:
1) Install a VPN client and connect to a VPN service. This creates an encrypted tunnel between you and the VPN server. You company cannot monitor traffic over a VPN connection.
2) Browse over Tor. Similar to a VPN service, Tor routes your traffic in a way that prevents the local network from snooping on the content you are browsing.
3) Use your phone as a mobile hotspot. Secure this hotspot using a WPA2 passcode. Your company can't eavesdrop on (2|3|4)G connections as that data is encrypted between you and your ISP (in this case, your phone service provider).
With enough time and effort, someone can correlate your observable traffic and other traffic and figure out who you were and what you were doing, but that's not something your employer will spend time and money doing.