Solved at last! I believe that the final answer is:
KRONBORG or ELSINORE
The completed jigsaw looks like this:
(with different shades of red, dark grey and light grey to show distinctions between adjacent pieces)
Clearly we have here the words 'HOME LANGUAGE' on the background of the flag of Denmark. We will need to derive from the rest of this picture a word or phrase which has some relevance in the Danish tongue...
How can we find these words?
Let's first highlight the tabs between each piece (shown here in blue):
With the hints in mind we now see that in the three central 'pixels' of each tile edge that jigsaws into another tile we have:
Numbers written in ternary. Thanks to a prompt from the OP in comments we know we must read them by orientating each side towards the top before inferring a number from it. In this way we read the following:
Right-hand edges of top row:
010 / 001 / 201 = 3 / 1 / 19 in base-10 = C / A / S in A1Z26
Bottom edges of top row:
202 / 110 / 012 = 20 / 12 / 5 in base-10 = T / L / E in A1Z26
Right-hand edges of centre row:
002 / 002 / 120 = 2 / 2 / 15 in base-10 = B / B / O in A1Z26
Bottom edges of centre row:
200 / 122 / 120 / 202 = 18 / 14 / 15 / 20 in base-10 = R / N / O / T in A1Z26
Right-hand edges of bottom row:
002 / 002 = 2 / 2 in base-10 = B / B in A1Z26
Taken altogether, this reads: CASTLEBBORNOTBB.
So what does this mean? Well, realise that:
This is the phrase 'CASTLE BB OR NOT BB', or reading 'BB' as '2B' - or rather, 'to be' - it could be read as 'Castle, "To be, or not to be"'. Since "To be, or not to be" is a famous quote from the Shakespeare play Hamlet, what we are ultimately being asked for here is the name of the castle in the play, which after all is set in Denmark.
In the play itself, the castle is known as Elsinore. However, in Danish (Hamlet's home language) it is more widely known as Kronborg.
This satisfies the title since:
A castle from a Shakespeare play could certainly be considered a 'big playhouse'!