[I'd paused my solve overnight, and the browser didn't tell me other solutions had been posted when I woke the PC back up - it was later determined that I'd had the somewhat unique approach of completely ignoring one of the rules making for a few detours from the intended solving path until I finally needed to invoke the missing rule at the end]
A solution that seems to fulfil all of the clues given:
I initially tried to use @bobble's Penpa link, but didn't really get on with the interface, so returning to MS Paint, I got most of the way to an incorrect solution...
At which point I realised I'd messed up... both of the '1 1 2' clues aren't met, and the remaining greens are unresolveable...
Using the experience from failing to solve it once, I started again from an empty grid...
I'm using the following colour scheme:
[dark] red/blue : this colour exclusively
[pale] red/blue : either the corresponding colour or possibly purple.
Purple : definitively known to be purple.
[pale] green : NOT purple (may be red or blue)
white/empty : could be anything
coloured blobs : identifying one of a group - especially in cases where one must be purple.
coloured lines : at least one of the cells either side of the line must be the specified colour.
First we can mark the really obvious stuff from the Tapa clues themselves, and the adjacent squares where two of them are close or one is near the edge and sufficiently restricted:
Next, look at the clues for the "Cross the streams" puzzle:
The '4' in column 5 is placeable immediately, several other columns/rows cause some cells to be known red, and others to have a known red adjacent to a given grid line.
This also allows a few more blues to be completed to avoid a 2x2 region of red. The "* 6 *" clue is particularly annoying - as either * could represent nothing at all, the 6 could be precisely either half of the grid so it gives us nothing at this stage.
At this point row 11 is particularly significant:
one of the "outer" reds must be connected to the "middle" red, because there are only 2 red regions. This must overlap the blue, and so gives us a choice of two locations for a purple, both of which "see" the two squares above the right hand possibility:
Next, observe in column 4
There must be 3 adjacent red cells, at least one of which must be adjacent to the '6' Tapa clue in column 5. It already has a red cell adjacent to it, and they cannot both avoid overlapping with blue. Indeed, there's only one place to put that '3' at the top of the column to avoid having more than 1 overlap with blue, so this gives us another purple that must be in columns 4 or 6.
Having a kind-of 'x-wing' formation of purples, plenty of other cells can be marked as "not purple" (or definitely red/blue if already known which), row 12 has two red regions identified, and needs the right-hand "red" to be able to connect, and a few other simple deductions to:
In the above image, observe that the cells marked with green ovals
are the balance of the bottom 4 rows - the starbattle must have 4 purple cells in the bottom 4 rows, and 4 purple cells in the bottom 4 regions, so one of the upper "green oval" cells is purple if and only if the lower one is.
However, the lower "green oval" cell sees both remaining possible locations for a purple in column 5 (empty "pink" cells above), so in fact all the green-oval cells must be non-purple.
Also observe in row 6 that one of the "pink" cells must also be purple to allow for 5 blues connected around the '5' Tapa clue above it.
Still in row 6, the "3 3" Tapa clue allows for only two directly opposing non-blue cells, so in particular one of these must be the red for the final '?' of the row 6 clue.
The Tapa clue in row 9 has a definite red, to the left, so the other non-blue must be either above or below it, with all remaining cells being at least blue.
Row 12, the purple must be in the right-hand region (this has been sitting here for a while...!), which means the purple on row 11 must be in column 4, also disambiguating some other purples... the immediate effect of this deduction (after marking remaining cells near a known purple as "not-purple") is:
"Cross the streams" row 11 gives us a definite red next to the Tapa 6, so we can complete the positions of blues next to that, which also fixes the Tapa 3 on row 9.
"Cross the streams" row 10 is then somewhat restricted,
more importantly, after marking as definite blue the cells which prevent a 2x2 red region near the Tapa 3 and 6, there are only two cells in the lower-left region that could possibly be purple, giving only two possibilities for the starbattle in rows 1 and 3:
We can then disambiguate
which side of the Tapa 5 in row 5 the blue cells are (they are to the left), because of the red in column 3 that became a definite red due to not being one of the two possible positions for a purple...
After ensuring connectivity of reds in the left-hand side without making a 2x2 square, and also respecting the '* 4 4 *' clue in column 3, we reach the following:
The top-left section is easily resolved
as the other alternative for connecting to the red would break the '3' tapa clue.
We can also resolve several cells around the '6' tapa clue in row 4, and note that row 5's purple cell is now restricted to one of 3 positions in the same region.
Considering the tapa clues with "1 1 2", these must have at least one red in each adjacent column and row. In particular column 8 with it's "2 1" cross the streams clue must have a red adjacent to the lower "1 1 2" tapa clue, so the red starting on the upper "1 1 2" tapa clue must be of length 2, and there must be no other reds in the column... this also disambiguates the '7' clue for the top row:
Row 7 is also much more restricted, and in particular gives a red in column 7, which reduces the possible positions of the 6 connected reds from the cross-the-streams clue from "* 6 *"
After marking the two possible positions for the purple in the right-of-centre region at the top, it's noticeable that each remaining purple is restricted to a single row within a single region, but none seem definite yet...
It was also around here I went wrong on my first attempt, failing to notice the possibility of a purple just below the upper "1 1 2" Tapa clue.
the "1 5" tapa clue needs the purple to complete the group of 5 blues.
We cannot have purple in row 12 column 8, as the "2 1" clue in column 8 would make 3 "blues" to the left of the "1 1 2" clue in row 11, so the purple in column 8 must be in row 5.
For the "3 3" Tapa clue, if the non-blue cells are diagonal, this would require a purple cell in row 4 column 11. This quickly leads to a contradiction, as the top right becomes fully enclosed in blue so the red cannot connect... so the non-blue cells must be vertical.
At this point, the only possibility for a purple
in column 11 is in row 7, just above the "7" Tapa clue, which must then be surrounded mostly by blue, but can connect left or down with a red if needed (or otherwise place a red anywhere "out of the way"!)
Considering the positioning of the red near the bottom of column 8, there seem to be two possibilities - exploring the red in the lowest position, this forces this layout...
Attempting to connect the disconnected red regions
the only ways legally possible whilst respecting the clues in the vicinity, I reached this near-solution, which fails due to not respecting the "? ? ?" clue in column 9 (variations to make it respect that clue will break others) nor indeed the "? ?" clue in column 12 (this false solution was "forced" by row clues and tapa clues).
Instead we should
place the red in column 8 at row 11, leading to the following, which allows more flexibility because there are fewer disconnected red regions:
Looking at row 3
the red region represented by the second '?' needs to connect to column 11 or to row 4 column 10 in order to come back down to the remainder. This gives us a 2x2 block which must contain a pure blue and a purple above the 2 reds.
Considering the Tapa clue 7 in column 11, this needs to connect either to the reds to its left or the reds below. Either way we get several blues from this...
The '3' in row 10 cannot be at the right, as this would disconnect the red regions, so it must be to the left, so we reach the following:
One way to proceed from here could be:
To loop the red into the right column and also have a red in the bottom-right to fulfil the "? ?" clue in column 12, however due to the two '4' in tapa clues in column 11, this forces 2 purples into column 10 which isn't allowed!
If we choose the bottom-right square to be
blue, the remaining purples are forced, and in order to fulfil the "? ?" for column 12, we need to do this:
at which point the not-purple in row 3 cannot be blue as it would break the tapa clue and cannot be red as it would create a 2x2 block.
Instead, the following layout seems to fulfil all of the clues:
reached by making the bottom-right square red, which forces the positions of all remaining purples, then choosing a layout that gets a "1 4" tapa clue fulfilled and connecting everything together.
The puzzle does not seem to have a unique solution - the pair of "not purple" in row 2 must contain at least one blue, but can otherwise be freely chosen without breaking any clues, and the other "not purple" can be arbitrarily chosen as red or blue.
From these possibilities, I arbitrarily chose the most aesthetically pleasing to present as the final chosen solution at the top of this answer.
As pointed out by @deusovi, this was because
I'd completely missed the rule "Blue shaded cells should form a single orthogonally connected group; no 2×2 cell area consists entirely of cells shaded blue.", and done the entire solve without relying on that rule at all...
therefore the presented solution is indeed unique as others had found before I saw their solutions.