"so as a consequence" is correct and usable in written English.
You can use it when explaining why something happened due to an earlier event. For example, "The store was flooded, so as a consequence, it had to close down."
And so, as a consequence, we focus on reducing costs.
"So as a consequence I ordered a gun.
"We didn't have that information, and so as a consequence, we went on with it".
And so, as a consequence, I think it is important for us to go ahead and answer these questions.
"I think that the more they mature, the better you get to know them, so as a consequence they are easier to train," Stoute says.
The ties that used to bind are weaker now, and so, as a consequence, are the politicians whom we elect, and the governments they form.
Some of the eyewitnesses gave evidence at Mair's trial, but one became so unwell as a consequence of what she saw that she could not attend.
Shafilea fell so ill as a consequence that she had to be rushed back to hospital in England where her weight plummeted to five stone.
Improvements in the social position of the peasantry did occur, but not so much as a consequence of the revolt as of changes in the economy that would have occurred anyhow.
We saw CARLY JIBSON, who is 19 and making her Broadway debut as the latest TRACY TURNBLAD, as well as THOMAS MEEHAN, who co-wrote the books for "Hairspray" and "The Producers," and is so rich as a consequence that he was getting his martinis at the cash bar.
Yoshida pledged that "the Government of Japan does not consider that the Government of the Netherlands by signing the Treaty has itself expropriated the private claims of its nationals so that, as a consequence thereof, after the Treaty comes into force these claims would be non-existent".
