Where is the real base for MI5?

The Headquarters of MI5 is Thames House, Millbank, Central London. The Service also certainly maintains numerous other discreet "facilities" ("safe house", secure locations etc) in and around Greater London, and probably in many other UK cities.

MI5 regularly uses training facilities (either MOD or Police controlled) in Kent, the Greater London & Home Counties areas, and also those situated in certain C19th fortifications around Portsmouth.

It is also very likely that there's a strong MI5 presence at GCHQ, Cheltenham, Gloucestershire, and at the Joint Services Intelligence Centre, Chicksands.

There's no secret "real base" for MI5: its HQ is in Central London in Thames House, but its personnel certainly regularly work in and out of numerous other locations - who knows where?!