Could Jesus have sweated actual blood in the garden of Gethsemane? In John 19 verses 33-36, it is written that 


  The Jews therefore, because it was the preparation, that the bodies should not remain upon the cross on the sabbath day, (for that sabbath day was an high day,) besought Pilate that their legs might be broken, and that they might be taken away. Then came the soldiers, and brake the legs of the first, and of the other which was crucified with him. But when they came to Jesus, and saw that he was dead already, they brake not his legs: But one of the soldiers with a spear pierced his side, and forthwith came there out blood and water. And he that saw it bare record, and his record is true: and he knoweth that he saith true, that ye might believe. For these things were done, that the scripture should be fulfilled, A bone of him shall not be broken


Modern medical experts explain the "blood and water" to actually be separated hemoglobin and plasma, the two major constituents of whole blood, which separate after death.  



Then Luke, a physician himself, wrote in his gospel account, chapter 22, verse 44:


  And being in an agony he prayed more earnestly: and his sweat became as great drops of blood falling down to the ground.


Key words:  "egeneto ho hidros ... haimotos" - "became the sweat ... blood" in the Interlinear.

So the question is in the title:  Could Jesus have sweated actual blood in the garden of Gethsemane?
Yes. Modern medical knowledge confirms the condition and the cause reported by the physician Luke using the word egeneto to describe how sweat became blood.

Turns out that recently the National Institutes of Health issued this report:

A Curious Case of Sweating Blood  (ncbi.nlm.nih.gov)

Excerpt:


  Hematohidrosis is a very rare condition in which an individual sweats blood. It may occur in an individual who is suffering from extreme levels of stress. Various causative factors have been suggested like component of systemic disease, vicarious menstruation, excessive exertion, psychogenic, and unknown causes. Fear and intense mental contemplation are the most frequent causes.


Taken in context it is clear he was certainly in "intense mental contemplation."



In studying the Interlinear of this section, the answer seems to be that his sweat becomes blood drops:

Luke 22 Interlinear



Written by the physician Luke, "egeneto ho hidros ... hiamotos" - "became the sweat...blood"