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You might be confusing interface MTU, 1514, with a protocol (IP) specific MTU, 1476.
As noted by Harold, the tunnel has defaulted to an IP MTU of 1476 (1500 less 24 GRE overhead), which can be the ideal size for a GRE tunnel. (Is any thing else shown for show interface #, like "Tunnel protocol/transport GRE/IP"?)
Since your IP MTU showed as 1476, then tunnel is probably using GRE and is providing a IP MTU that's 1500 less GRE header of 24, i.e. 1500 - 24 = 1476.
The 1514 MTU would be the MTU the tunnel is providing. i.e. "What is the actual MTU applicable to the interface."
I haven't tried this myself, but since the tunnel's MTU isn't a real physical MTU, I would assume it can be increased using the MTU command. NB: having the MTU larger than what the path's physical MTU can support will require the packet to be fragmented to forwarded.
